Equations of Conic Sections

Welcome to our video lesson for today. Now we will discuss about the equations of conic sections. First, let's have our lesson objectives. First is to identify the key components of the graph of a conic section. Second, determine the general and standard equation of conic sections. And plus, we're going to transform the equation of conic sections from general to standard form and from standard form to general form of equation.

Now let's begin with the parts of the graph of conic sections. In this part, we are going to specify some parts of the graph of conic sections. Then we will, uh, elaborate the different parts of the conic sections when we are discussing the topic about graphic exceptions. Again, for this part, some parts lamang po yung idi-discuss natin. So, start tayo with circle. In circle, we have two main components that we needed. First is the radius. So, we know that the radius is a line that connects the center of the circle and any point of the circle. So that is the radius. Now, the second part is the center. When we say center of the circle, it could be on the origin. Okay? When we say on the origin, nasa origin coordinate ang ating uh center. And also, pwede rin namang ma-move yung center ng ating circle. Pwede rin siyang mag, ah, wala sa origin, nasa HK coordinate. When we say HK coordinate, nasa X and Y coordinate siya, wala sa origin. Okay? So, those are the two main components na kakailanganin natin pag in-solving the equations of circle.

Next, we have the ellipse. For ellipse, there are three variables or three main components that we needed for solving the equation or transforming equations ng isang ellipse. We have the A variable, the B variable, and the C variable. For A variable, it is the line that connects the center of the ellipse. Okay, this one is the center of the ellipse and this is the vertex of the ellipse. Okay. So, that is our A and A variable or A value is the longest line among this other variable which is the B and C. When we say B variable naman, it is the line or the measurement from the center of the ellipse after to the C vertex. And for C, we have, uh, it is the line from, uh, from the center of the ellipse to the focus or F of the ellipse. So, those are the three variables that we needed for the transforming equations of ellipse.

Next, we have this hyperbola. For hyperbola, we also have A, B, and C variable. Wherein, um, A is the length of transverse axis. Okay? Ito yun yung red one. B is the conjugate axis and C is the transverse axis. Okay? So, on this one, ito naman yung kakailanganin nating mga variables and values that we needed for transforming equations of hyperbola. Okay.

And last, for parabola, we have the C. Okay. C is the measurement from the focus of the vertex to the focus of, sorry, from the focus of the parabola after the vertex of the parabola. And, uh, the point from here to here, the measurement of this must be 4C. 4 times the length of the C value again. So, those are the points or key points that we needed in solving and transforming equations of each type of conic section. Okay.

Next. So, before we proceed, no, uh, in transforming equations, we must be familiar with the different types of equations or different form of equations ng type of conic section. So, first, we have a circle. Tulad ng na-mention ko kanina, circle, ang center niya pwedeng nasa origin. So, this will be an example, no, of a circle with the center in the origin. And also, pwede rin na ang circle ang center niya ay wala sa origin. Example ito. Ito yung origin natin, but wala diyan yung center. This one nasa HK siya. So, magkaiba ang ang general and standard form ng circle kapag nasa origin ang center and kapag nasa HK coordinate ang center ng circle. So, for circle, we have this general form x² + y² - r² = 0. And, uh, its standard form, we have x² + y² = r². So, as we can see from general to standard form, we transpose the value of our radius. Again, this general and standard form of equation is for circle with an origin of in the center.

Next, proceed tayo with a circle with the center at HK. So, we also have the general form and the standard form. For general form, we have x² + y² + dx + ey + f = 0. And we know, yung general form ng circle is dapat equal or present muna yung x² and y². Dapat they have same sign and also they must have same coefficient. And for the standard form, from this general form, we are going to transform it into standard form and this will be the standard form of circle. We have (x - h)² + (y - k)² = r². Okay? So, this one naman yung general form and standard form again kapag ang circle ang origin ay nasa HK. Okay.

Next, we have ellipse. Yung ellipse natin pwedeng maiba yung orientation niya or yung position, no. When we say ellipse, pwede rin siyang origin ay nasa zero. Ah, sorry, ah, pwede rin yung center ay nasa zero. Pwede rin naman na nasa HK. Okay? And also, pagdating sa orientation ng ating ellipse, pwede siyang, ah, horizontal. This one is horizontal ellipse and the other one is the vertical ellipse. Okay? So, these are the types of ellipse that we can, uh, form, no, or graph. So, unahin natin yung origin, ah, ellipse na ang center ay nasa origin. So, what will be the general form and standard form? Okay. So, first, we have this ax² + cy² + f = 0. Take note that A must be less than C. And for standard form, we have x²/a² + y²/b² = 1. A on this one, A must be greater than B. For us na ang center ay nasa HK, we have this general form ax² + cy² + dx + ey + f = 0. And its standard form is (x - h)²/a² + (y - k)²/b² = 1. So, tandaan, no, ang ellipse natin, laging ang operation natin between these two groups are always addition. Okay? Ayan.

Now, next, we have this one. Okay. The general form and standard form, and also the general form and standard form kapag ang center naman ay nasa HK coordinates. Okay? We have this one. So, as you can see, ang daming mga equations, no, na meron tayo dito. At once na mag-so-solve tayo, ay, madali niyong mano-maneuver. Now, ano ba ang difference between this two, uh, two rows ng ating ellipse? Ito yung nag-i-indicate ng orientation ng ellipse natin. It could be horizontal ellipse or vertical ellipse. So, nasa upper part or upper row natin, this is for, um, vertical, ah, sorry, horizontal ellipse. Okay? So, paano natin malalaman na horizontal ellipse yung meron tayo by just observing the given equation? Take note that this A variable, okay, this A variable, it is paired with x. It means, yung A natin, which is the longest, uh, line or values that we have, ay nasa x coordinate. Okay? So, it means it is horizontal. On the other one naman, A is paired on our y sub. It means, yung ating longest line ay nasa y coordinate. So, it is vertical ellipse. Okay? So, that is the difference between the two. And also, by just observing the given equation, madali nating maa-identify kung horizontal or vertical ellipse yung meron tayo. Okay?

Now, let's proceed with our hyperbola. For hyperbola, pwede rin siyang vertical or horizontal hyperbola. Okay? So, yung hyperbola, pwede rin tayong magkaroon ng center na origin tsaka sa H. And what are those, uh, equation? We have this general form and standard form. For standard form of our hyperbola, we have here x²/a² - y²/b² = 1. And for the general form of hyperbola at HK, we have ax² - cy² + dx + ey + f = 0. And its standard form is (x - h)²/a² - (y - k)²/b² = 1. So, this one naman, it is horizontal. Okay? And the other one. Okay. So, we have here this general form and standard form. Again, um, almost same lang sila. Magkakaiba, magiging A difference lang between the two is yung magiging orientation. The other one, the other one is horizontal and the next one is for vertical. So, yung horizontal natin, yung A ay nasa x and yung, ah, for vertical, yung A naman ay nasa kay Y axis natin or Y coordinates. Okay.

Next, we have for parabola. For parabola, maraming, ah, equations or maraming form of equations yung meron. Okay? Bakit marami? Kasi yung parabola, pwede siyang magkaroon ng apat na, ah, apat na possible opening. It could be opens upward, opens downward, opens to the left, and opens to the right. Okay? So, almost same lang lahat ng ating mga equation dito. Nagkaiba lang yung pagdating sa, syempre, kapag, ah, opens to the right or left, y² yung nasa left side. Okay? And yung nasa right side yung ating 4cx. Okay? Or -4cx. Pagdating naman sa standard form, y - k², standard form ng HK. Nasa HK naman ang, ah, vertex ng ating parabola. This case, we have the standard form of (y - k)² = 4c(x - h). So, again, this equations, ah, magsasabi, no, kung ano ang magiging opening natin ng, ah, parabola. Kapag positive, it's either opens to the right or upward. Kapag naman negative, it's either, um, opens to the left or downwards. Okay?

Now, to summarize again, these are the different general form or standard form of equations ng conic section. Tandaan, magkaiba ang standard form at ang general form ng center or vertex, no, ng nasa origin and kapag naman nasa H4. Now, since we are already familiar with the different, um, form of equations ng type ng each type of conic section, now let's proceed with transforming equations from general to standard form. Okay. On this one, we are going to use the, um, the completing the square for us to be able to transform equations from general to standard form. So, here are guidelines. We are going to use alter, alter, a, we're going to assign ex on this one. Sa general form of equation pa lang, we can already identify what type of it is, okay, using the flow chart na na-discuss natin before. Second is to locate the F term to the right hand side of the equation. So, yung F term natin, yung constant, Okay, we're going to, um, locate it or transpose it into the right side. Number three, team X and Y. It means we're going to regroup our equations from the left side. No, lahat ng X magkakasama, in X variable and lahat ng Y variables ay magkakasama in different groups. Number four, we are going to execute completing the square. Okay. And for last, number five, we are going to recheck the result using step number one. So, it means after, uh, completing the square, we can already, uh, make the general equation into standard form. Then, um, by observing the given, uh, standard form, we can also, uh, determine what type of conic section yung, so it means kung sa general form ellipse yung na-observe natin using the equation, dapat at the end, after nating mag-transform, equation ellipse pa rin dapat yung, hindi dapat magbabago yung type of conic section once na mag-transform tayo ng equations from general to standard form. Okay.

Now, so again, these are the guidelines. Let's apply this. Okay. So, let's have example number one. For example number one, we have the given x² + y² - 4x - 8y + 16 = 0. Now, step number one, we are going to identify what type of conic section it is. So, this one is circle. Bakit circle? Kasi yung x² and y² natin, they have similar sign or same sign and also they have same coefficient. So, it means it is a circle. Now, let's apply step number two, wherein we are going to transpose iyung ating F term or iyung constant. So, it will be x² + y² - 4x - 8y = +16 will be -16. And after that, we are going to regroup. Group natin yung same variable. We have x² kasama niya yung -4x. Then plus, we're going to allow space for us to perform completing the square. Now, um, this one, next is plus. I-group naman natin yung ating y variable. So, we have y² - 8y plus, allow ulit ng space for completing the square = -16. Okay. Next, perform tayo ng completing the square. On this one, we are going to use again the B/2 squared formula for us to be able to, um, for us to determine or solve for the third term of the equation. Okay. So, on this one, our B is -4. So, -4/2 squared. Sa kabila naman sa y terms natin, our, uh, y, ah, sorry, our B is -8/2 squared. Then perform the operation. We have x² - 4x + (-4/2)² = x² - 4x + 4. Plus, we have y² - 8y + (-8/2)² = y² - 8y + 16. Equals to -16. Okay. So, this time, nag-add tayo sa left side ng 4. Then sa Y group natin, mag-add tayo ng 16. Then perform completing the square. Ah, sorry, this one is perfect square trinomial. So, we need to change it into square of binomial. So, it is (x - 2)² + (y - 4)² = -16 + 4 + 16. Okay. So, -16 + 4 + 16 = 4. So, it means four na lang yung nasa right side. This one is the standard form of our equation. So, we have (x - 2)² + (y - 4)² = 4. So, again, this is the standard form of the equation. Now, we need to identify the different components of the circle. Sa, sa circle naman, no, kailangan natin mahanap yung center and the radius, no? Center natin, since we have here a form of x - h² + y - k² = r². So, una yung radius natin will be, since yung 4 natin naka-squared pa yan, so we need to get the square root. So, the square root of 4 is 2. Okay. So, ulitin natin. Pakita natin yung process. Again, R natin is 4. Since naka-square yung 4 or yung radius natin, so kukuha na natin yung square root of 4, which is, it will give us a result of 2. Next, for our, ah, center naman, since wala sa origin, kukuhanin natin yung value ng H and K. Take note that yung H is laging kasama ni X and yung K naman ang laging kasama ni Y. Okay. Dito sa pagkuha ng H and K, yung opposite sign ang kukunin natin. So, since this one is x - 2, negative siya dito. Gagawin natin siyang 2. For K naman, we have here y - 4. Negative yung 4 natin. So, it will be positive 4. Again, opposite ng sign yung kukunin natin or gagamitin natin for our H and K values. Okay. So, yun, we're done with, um, identifying transforming equations from general form to standard form and we are also done with solving the different parts of our circle. We have the radius, the HK. So, therefore, our center is at, nasaan ang ating center? Nasa 2, 4 coordinates. Okay? It means kapag nag-plot tayo ng graph, dapat ang center natin ay nasa 2, 4 coordinates. Okay? So, that is our example number one. Okay?

Now, let's proceed with our example number two. For our example number two, we have x² + 3y² + 8x - 12y + 10 = 0. Again, we are going to identify the type of conic section. So, on this one, they are both positive. Positive x² and 3y², but they have different sign, ah, sorry, different coefficient. So, therefore, it is an ellipse. Okay? Now, perform ulit tayo ng transforming of equation. So, it means yung ating constant, ita-transpose sa right side. So, it will be x² + 3y² + 8x - 12y = to, yung +10 will be -10. Okay? Then after that, regroup tayo. Group natin yung x² + 8x. Maglagay ng space for completing the square. Then plus, we have for the second group, the y variable naman. So, that would be 3y² - 12y plus, allow ulit ng space for completing the square = -10. So, this time, so since wala namang coefficient yung first term natin kay x, yung x², means we can already perform completing the square. So, it will be, it will be x² + 8x. So, this one, no, para mahanap yung third term, again, ang gagamitin lang natin ay yung B/2 squared na formula. So, that will be our B is 8, then squared. So, it will be 8/2 = 4, that is 16. Plus, next, on this one, no, merong coefficient yung ating y², which is 3. So, it means pwede tayo dito mag-factor out. Mag-factoring, factor out natin yung 3. 3y²/3 = y². And -12y/3 = -4y. So, therefore, on this one, our B term is -4, and divide it by 2, then perform the operations. 3(y² - 4y) plus, allow ulit ng space for completing the square = -10. So, para mahanap yung third term, -4/2 = -2, squared is 4. So, 3(y² - 4y + 4) = -10. Now, since sa left side ng group ng x variables natin, we added 16. So, maga-add tayo sa right side ng 16. Then for the, uh, group of y variables, nag-add tayo ng 4, but this 4 is multiplied by 3. So, therefore, maga-add tayo dito ng 3 * 4 = 12. Okay. Next, this time, we already have here perfect square trinomial. So, we need to change it into square of binomial. So, it will be, get the square root of x², that is x. Get the square root of 16, that is 4. Then for the operation, the operation of second term, okay. Plus 3 times, um, yung perfect square trinomial natin na y variables, make it, um, square of binomial din. So, square root of y term is y. Square root of 4 is 2. For the operation, since yung, ah, second term natin is subtraction, so subtraction din yung gagamitin natin as the operation for our square of binomial. Then equals, simplify lang natin to. -10 + 16 = 6. + 3 * 4 = 12. Ah, 3 * 4 is 12. So, + 6 + 12 = 18. Okay. Now, on this one, although we are done with our with the completing the square process, but for an ellipse, para maging isa si standard form, dapat ang ating right side of the equation ay equals to 1. Now, paano ba natin gagawin equals to 1 yung right side of equation? Since mayroon tayo dito, 18, mag-divide lang tayo on both sides by 18. Okay. So, therefore, it will be, so wala naman tayong maka-cancel dito sa x group natin. It will be (x + 4)² / 18 + 3 * (y - 2)² / 18 = 18 / 18 = 1. Then after that, no, ah, sorry, so nag-divide tayo ng 18, ah, sa ating left side. Now, as you can see, as you can see here, no, sa ating y, meron tayong pwedeng ma-cancel out, yung 3 and 18. So, yung 3 magiging 1 kasi 3 divided, we're going to divide it by, um, their greatest common factor, which is the 3. No? So, idi-divide natin yan. So, it will be one na lang and yung 18 will be 6. Through, through cancellation method, no, we can simplify this term. Ayan. Therefore, we have (x + 4)² / 18 + (y - 2)² / 6 = 1. That is the standard form of equations that we have. Okay. So, from general form to standard form, and after that, no, once na ma-change natin into standard form, hahanapin naman natin yung different values that we needed for graphing naman. Okay. So, on this one, kailangan natin yung variable H, variable K, variable A, variable B, and variable C. For our H and K, again, H lagi siyang kasama ni X variable. So, it means nandito siya. Okay, this is our H. But opposite yung opposite ng sign yung kukunin. Since ito ay +4, pagdating sa ating H, it is -4. Okay. Next, for Y, pair niya lagi ay si K. So, since -2 yung nasa equation natin, kukunin yung opposite sign, it will be positive. Need to write the positive. Okay, it will be positive 2. Next, for our A. Okay. Tandaan ang ellipse natin, ang A laging yung mas malaking value sa denominator. So, since we have 18 and 6, para hindi tayo mahirapang mag-identify ng A and B, so alin ang mas malaking value, which is 18. So, take note that our denominator ay naka-square din po yan. So, therefore, to get the value of A, get the square root of 18. Plus, ah, sorry, get the square root of 18. So, that is, solve natin, 4.25. So, that is approximately, approximate is 4.24. Our B, so, natin yung square root of 6. That is approximately, so ito yung sign natin, no, pag nag-a-approximate tayo. Yan, approximately 2.445. Okay. Now, let's proceed with solving for C value natin. For C value, we have a formula of square root of, square root of A squared minus B squared. Okay. So, wherein our A squared is 18, then minus B squared is 6. Get the square root of 12. So, that is approximately, solve natin, the square root of 12. So, that is equal to 3.46. So, approximately 3.46. Okay. So, this is our value of C, value of B, value of A, K, and D, H. So, those are the important, um, variables na kailangan nating mahanap yung values. Okay. Again, this values of HK, A, B, and C ay gagamitin natin for graphing. Okay. So, that is for our example number two.

Now, let's proceed with our example number 3. So, for example number 3, we have the given y² - 5x² + 30x + 4y - 46 = 0. So, same process yung gagawin. Transpose muna. We have y² - 5x² + 30x + 4y = so, -46 will be positive 46. And after that, regroup tayo. So, on this one, group natin yung y² plus, um, kasama niya si, si +4y, then allow ng space for completing the square. Then plus, okay, plus, ah, sorry, it is minus. Okay, minus. So, 5x² group naman niya ay si 30x, plus, allow ng space for group, ah, for completing the square, then equals to 46. Now, bakit nga ba to naging subtraction? Okay, bakit naging minus? Now, if we're going to observe on the given equation, we have y² - 5x². And on this one, by observing this two terms, malalaman natin that they have different sign. It means it is hyperbola. And for the general form or and standard form of hyperbola, the operation is subtraction. Okay? Kaya ito ay subtraction. Unlike kay ellipse, laging addition yung operation, but for hyperbola, subtraction po yan. And also, if you're going to notice on this one, negative din dito, so subtraction siya. But again, no, uh, the best way para malaman kung ano yung operation at hindi tayo malito sa pag-identify pa lang ng conic type ng conic section using the general form, at nalaman na hyperbola to. So, automatic, no, it is subtraction. Now, after that, um, perform tayo ng completing the square. To solve for our third term, same pa rin yung gagamitin, B/2 squared. On this one, our B is 4. So, it will be 4/2 squared. On the right side, we have 5x² + 30x. Before tayo mag-perform or hanapin yung ating third term, since yung 5 and 30 ay may common factor, so magfa-factoring muna tayo. Divide both terms by 5. Maka-cancel out yung 5. 30 magiging 6. So, therefore, uh, our third term, ah, second term or B term natin dito ay 6. So, therefore, the third term is 6/2 squared. And perform the operation. We have y² + 4y + (4/2)² = y² + 4y + 4. Minus, ah, sorry, 5 times, kasi nag-factor out tayo, x² + 6x + (6/2)² = x² + 6x + 9. Equals to 46. So, left side yung y group natin, nag-add tayo ng 4. So, it means sa right side, maga-add tayo ng 4. Then for x groups natin, nag-add tayo ng, but we know that it is multiplied by -5. So, therefore, ima-multiply na mag-ma-ma-ga-add tayo ng -5 * 9 = -45. Okay. Next. Then make it square, uh, square of binomial. So, square root of y² is y. Square root of 4 is 2. Then the operation of second term is addition. Okay. Minus, 5 times, square of binomial again. Square root of x² is x. Square root of 9 is 3. Operation of second term is addition. Okay. Meron tayo sa left side na (y + 2)² - 5 * (x + 3)² = to, para hindi mahirapan, simplify muna ng -5 * 9 = -45. Okay. So, natira is, ah, 46 - 45 = 1. Plus 4 = 5. And this one, since hyperbola to, dapat ang goal natin, ang right side ng equation is one lamang. So, to make it one, need natin mag-divide on both sides by 5. Again, for ellipse and hyperbola, ang goal natin palagi is maging one yung ating right side. Therefore, we have here (y + 2)² / 5 - 5 * (x + 3)² / 5 = 5 / 5 = 1. Then, cancel out 5/5. So, magiging x + 3 squared. Okay. 1. So, wala na yan kasi ang denominator natin is 1. Equals to. And this is the general form, ah, sorry, the standard form of the given equation of a hyperbola. Okay? So, naka-standard form na to. Then after that, syempre, we need to solve for the different, uh, values that we needed. Kailangan natin si H, si K, dahil wala naman siya sa origin, no, nasa HK siya. We also need A. We also need B. Also need C variable. For our H, again, kasama lagi siya ni X. So, therefore, ang H natin ay ito yung 3. And get the opposite oper, opposite sign. Since positive 3 yan, it will be -3. For K naman, lagi siyang ka-pair ni Y. So, therefore, this is our K. So, y + 2, positive yung sign nito. So, therefore, negative naman yung kukunin natin yung value ni K. For A and B, no, baka malito lang kayo. Tandaan for hyperbola, ang A naman natin ay hindi A and B ay hindi magma-matter kung sino yung mas malaking value. For hyperbola, magma-matter naman dito ay kung sino yung positive. Okay. This one, meron siyang denominator na one. Alin ba ang group na positive dito? Si Y or si X? So, ang positive natin dito ay si Y. Therefore, nandito yung ating A, which is square root of 5. Okay. Now, solve natin yung square root of 5. That is approximately 2.24. Okay. Again, it is approximately 2.24. For our B, square root of 1. So, syempre, it is equals to 1. And for C, ayan, huwag malilito doon sa ating pagkuha ng C, no. Magkaiba yung pagkuha ng C kay ellipse and pagkuha ng C kay hyperbola. Kay ellipse ay subtraction. Kay hyperbola naman is we're going to use the formula A squared plus B squared. So, that will be square root of A squared plus B squared. So, that is square root of 5 + 1. So, approximately, so solve muna natin, no. Find the square root of 5 + 1. So, that is approximately, square root of 6. So, that is approximately 2.45. Okay? So, this is our C. So, on this example, tapos na tayo, no? Tapos na tayo. Nagawa na natin yung goal natin, which is to transform general form of hyperbola to standard, and also we already solved for the different variables that we needed for graphing of this equation. Okay.

Next, let's proceed with example number four. Sorry, four na to. Ayan. So, we have x² - 2x + y + 8 = 0. So, this one, same process lang, no? Transpose, um, sorry, hindi siya magiging same process. Okay? Um, para hindi tayo magkamali, identify natin yung type ng, uh, conic section. So, on this one, ang present lang ay x². It means walang y². Isa lang ang present na variable, uh, na may, na naka-square. So, it means it is parabola. Okay? Sa parabola, sa parabola, um, ang goal naman natin dito is maiwan yung, yung variable na naka-square. It means igu-group natin yan. Then lahat ng may, ng x variable nasa left side. That will be x² - 2x. Then allow space for completing the square. Equals yung y and positive 8 natin ay ita-transpose natin. So, it will be -y - 8. Then after that, perform tayo ng completing the square. So, it will be x² - 2x. To find for the third term, B/2 lang din po ang gagamitin natin. On this one, we have -2/2 = -1, squared that is +1. Equals -y - 8. Okay. Now, on this one, um, gawin na ulit natin yung perfect square trinomial. Gagawin na square of binomial. So, it is (x - 1)². Equals, ah, sorry. Before natin makalimutan, no, um, since nag-add tayo dito ng 1 sa left side, we need to add 1 din sa right side. So, it will be -y, then -8 + 1 = -7. Okay? So, this is the standard form of equation ng isang parabola. Okay? So, with that, no, ah, ito na yung process of solving the standard form or transforming the standard form of parabola. Okay.

Next, let's proceed with transforming standard, uh, transforming standard form to general form. So, we are done with transforming general to standard. This time naman, ibabalik natin, no, yung standard to general. So, there are several guidelines din or steps that we need to, um, follow. First is identify conic section. Again, yun naman yung mahalaga, no. Una pa lang, by observing the given equation, standard man yan or general equation, kaya naman natin i-identify. Second is expand the square of binomial. Third is use AP, or addition property of equality for F term. And four, arrange the terms according to the general form of the type of conic section, then simplify. And also, number six, recheck ulit natin kung ellipse siya sa standard form, dapat ang ma-transform yung equation, ellipse pa rin. Kung hyperbola siya sa start or sa general, dapat sa general, hyperbola pa rin. Okay.

So, let's have example number one. We have (x - 2)² + (y - 4)² = 2². Okay. So, this time, pabaligtad yung gagawin natin. Ito ay standard form. Ang goal ay maging general form. So, let's start with this. So, una, expand muna natin yung equation. So, pag-expand ng equation, we need to first is square the first term. This is our first term. So, square of x is x². For the second term, we need to multiply first term and second term. So, x * -2, that is -2x. Then imu-multiply natin to by 2. So, that is -2x * 2 is -4x. Okay. Again, ang process natin, yung first term and second term ipag-multiply and their product ay imu-multiply by, sorry, we're going to multiply it by 2. Always 2 ang imu-multiply natin. Okay. So, x * -2, that is -2x * 2, that is -4x. And last, no, we need to get the square of second term, square ni -2, which is positive 4. (-2)² is positive 4. Okay. So, yung ating square of binomial na-expand natin and nagawa nating perfect square trinomial. So, that is the first group natin kay x variable. Now, proceed tayo kay y variable. So, same process, get the square of first term. This is our first. So, y² is square of y². Next, yung y first term, -4 second term, ipag-multiply, which is y * -4, that is -4y. And the product must be multiplied by 2. So, it is -8y. And for the third term, get the square of second term natin dito. So, which is -4. The square of -4 is +16. Okay. Equals 2² is 4. Okay. 2² is 4. Then after that, um, eliminate natin yung grouping symbol, no? Tatanggalin na natin yung grouping symbol. So, it will be x² - 4x + 4 + y² - 8y + 16. Then yung constant natin sa right side, ita-transpose na natin sa left side. So, that will be -4 = 0. Okay. Then after that, regroup na natin. Okay, ah, sorry, rearrange na natin. Sorry, yan, rearrange. So, again, our form is ax² + bxy + cy² + dx + ey + f = 0. Okay. So, that is the general form ng isang conexion. But this one is circle. For may certain pattern tayong susundin. So, una, unahin natin sa x², followed by y², followed by x term or dx term natin. So, that is -4x. Okay? Followed by y, -8y. Then yung 4 and -4 combine na natin. 4 + (-4) is 0. And may another constant pa tayo dito. So, that is 0 + 16. So, +16 = 0. So, ito na ang ating general form from the given standard form of equation. Dito sa given equation, circle siya. Pagdating ng standard ng general form natin, it is still circle. It means correct ang process natin for for transforming this equation. Okay.

Next, let's proceed with example number two. For example number two, same process yung gagawin natin. Expand the terms. So, yung square of binomial expand natin. It will be square of first term. So, y² is y². Okay. Next, multiply yung first term and second term. y * -1, that is -y. Then multiply the product by 2. So, that is -2y. For the third term, um, get the square of this second term or square of 1, that will be +1. Then equals. On this one, um, simplify din natin. Distribute natin inside the parenthesis. So, that is 2 * x = 2x. 2 * -6 = -12. And after that, our goal is all terms must be placed on the left side and zero yung matira sa right side. So, it will be y² - 2y + 1. Then yung +2x will be -2x. -12 will be +12. Equals to 0. Okay. Then after that, rearrange lang natin yung terms and combine like terms. So, we have here y² - 2y - 2x. Then we have here 1 or positive 1 and +12. So, 12 + 1 = 13. So, +13 = 0. And with this, we're done with transforming equation. So, first given na standard form, we know that it is a parabola. Then the general form after nating mag-transform is still parabola. Okay.

Next, proceed tayo with another example. Example number 3. We have (y - 2)² / 25 + (x - 3)² / 5 = 1. Now, this one, so we have here operation of addition. They are both, uh, addition, so it means, sorry, positive, so it means it is ellipse. Okay? For, for ellipse or hyperbola, same lang naman ng magiging process, yan, no. Ang gagawin lang natin is una, eliminate the denominator. So, for us to eliminate the denominator, we need to multiply both sides of the equation by the LCD. And yung LCD ng 25 and 5 is 25. Okay? Multiply natin on both sides. Cancel out 25 on the first term. Okay, maka-cancel to. Matitira na lang is yung y² - 4y + 4. Plus, 25/5, so meron pa yang matitira or it has a quotient of 5. Again, 25 / 5 is 5. Then copy yung x - 3 squared. Equals to 1 * 25, that is 25. Again, the process on how we eliminated the denominator is by multiplying both sides by their LCD. Okay. Now, on this step, natanggal na natin yung denominator. So, it means kailangan na nating mag-expand ng terms. So, on this one, square of binomials. So, um, ganun ulit ang process. Paulit-ulit lang naman yung pag-expand, which is get the square of first term, which is y². Second, multiply yung first term and second term, which is y * -2, that is -2y. And their product must be multiplied by 2, which is -4y. For third term, get the square of second term, which is -2², that is 4. Okay. Plus, on this one, before nating i-distribute yung 5 na naka-factor, we need first to simplify or expand yung ating exponent. So, it means copy muna natin yung 5 sa labas ng group, then expand natin yung square of binomial. So, get the square of first term, x, ah, sorry, square of x is x². Then multiply first term and second term, that is x * -3, -3x. Their product must be multiplied by 2, -3x * 2, that is -6x. Then last, for the third term, get the square of -3, so square of -3 is 9. Equals to 25. Now, before nating ah mapag-combine yung like terms, on this part of the equation, meron pang 5 na nasa naka-factor or nasa labas ng grouping symbol. It means we need to distribute it first sa ating grouping symbol. So, it will be y² - 4y + 4 + 5 * x² = 5x². 5 * -6x = -30x. 5 * 9 = 45. Equals to 25. And after that, eliminate natin yung grouping symbols and combine like terms. Okay. So, it will be y² - 4y + 4 + 5x² - 30x + 45 = 25. Then after that, rearrange natin. Okay. Una palagi ay yung x², yung x² term ang first term, followed by the y², followed by x term, which is -30x, followed by y term, which is yung -4y. Followed by the constant, we have 4 and 45. 4 + 45 = 49. Okay. Then yung constant natin na 25, ita-transpose na natin on the left side. So, therefore, magiging, um, magiging +49 - 25 = 0. Okay. Then after that, ay, um, isi-simplify pa natin, no, kasi meron pa tayong two constant here. So, re-write lang natin. 5x² + y² - 30x - 4y. 49 - 25 is +24 = 0. And this is the general form of the given equation. Okay. So, with that, so that is the process and how we perform transforming equation from standard form to general form. And that's all for our lesson for this session. Okay. So, we're done with identifying the components of the graph of conic section. They are also done with transforming equations from standard form, from general to standard, and last, yung from standard to general form of equations. So, that's all for this session. If you have questions or clarifications, you can ask, no, the, uh, on our GC or by joining the, uh, online meeting. Okay. So, for now, that's all and thank you for watching this.