Answered

IDNLearn.com: Your trusted platform for finding reliable answers. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.

What is the coordinates of the center of an ellipse defined by the equation 16x^2 + 25y^2 + 160x - 200y + 400 = 0 ? Please give solution .

Sagot :

Olemakpaduidn

16x^2 + 25y^2 + 160x - 200y + 400 = 0     Rearrange and regroup.

(16x^2 + 160x) + (25y^2 - 200y ) = 0-400.     Group the xs together and the ys together.

16(X^2 + 10x) + 25(y^2-8y) = -400.     Factorising.

We are going to use completing the square method.

Coefficient of x in the first expression = 10.

Half of it = 1/2 * 10 = 5. (Note this value)

Square it = 5^2  = 25.     (Note this value)


Coefficient of y in the second expression = -8.

Half of it = 1/2 * -8 = -4. (Note this value)

Square it = (-4)^2  = 16. (Note this value)


We are going to carry out a manipulation of completing the square with the values

25 and 16.  By adding and substracting it.


16(X^2 + 10x) + 25(y^2-8y) = -400

16(X^2 + 10x + 25 -25) + 25(y^2-8y + 16 -16) = -400

Note that +25 - 25 = 0.    +16 -16 = 0. So the equation is not altered.

16(X^2 + 10x + 25) -16(25) + 25(y^2-8y + 16) -25(16) = -400


16(X^2 + 10x + 25) + 25(y^2-8y + 16)  = -400 +16(25) + 25(16)    Transferring the terms -16(25) and -25(16)

to other side of equation.  And 16*25 = 400


16(X^2 + 10x + 25) + 25(y^2-8y + 16)  = 25(16)


16(X^2 + 10x + 25) + 25(y^2-8y + 16)  = 400

We now complete the square by using the value when coefficient was halved.


16(x-5)^2 + 25(y-4)^2  = 400

Divide both sides of the equation by 400


(16(x-5)^2)/400 + (25(y-4)^2)/400  = 400/400              Note also that, 16*25 = 400.


((x-5)^2)/25 + ((y-4)^2)/16  = 1

((x-5)^2)/(5^2) + ((y-4)^2)/(4^2)  = 1


Comparing to the general format of an ellipse.

((x-h)^2)/(a^2) + ((y-k)^2)/(b^2)  = 1


Coordinates of the center = (h,k).

Comparing   with above   (x-5) = (x - h) , h = 5.

Comparing   with above   (y-k) = (y - k) , k = 4.

Therefore center = (h,k) = (5,4).

Sorry the answer came a little late.  Cheers.

Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.