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please can you solve this ellipse for me . 4y^2 - 8y + 25x^2 +150x - 171 = 0 with details and drawing

Sagot :

olemakpadu
4y^2 - 8y + 25x^2 +150x - 171 = 0

4y^2 - 8y + 25x^2 +150x - 171 = 0    Rearrange and regroup.

(25x^2 + 150x) + (4y^2 - 8y) = 0+171.     Group the xs together and the ys together.

25(X^2 + 6x) + 4(y^2-2y) = 171.     Factorising.

We are going to use completing the square method.

Coefficient of x in the first expression = 6.

Half of it = 1/2 * 6 = 3. (Note this value)

Square it = 3^2  = 9.     (Note this value)


Coefficient of y in the second expression = -2.

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

Square it = (-1)^2  = 1. (Note this value)


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

9 and 1.  By adding and substracting it.


25(X^2 + 6x) + 4(y^2-2y) = 171.

25(X^2 + 6x + 9 -9) + 4(y^2-2y + 1 -1) = 171

Note that +9 - 9 = 0.    +1 -1 = 0. So the equation is not altered.

25(X^2 + 6x + 9) -25(9) + 4(y^2-2y + 1) -4(1) = 171


25(X^2 + 6x + 9)  + 4(y^2-2y + 1) = 171+25(9) +4(1)   Transferring the terms -25(9) and -4(1)

to other side of equation. 


25(X^2 + 6x + 9)  + 4(y^2-2y + 1) = 171+25(9) +4(1)


25(X^2 + 6x + 9)  + 4(y^2-2y + 1)  = 400

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


25(x+3)^2 + 4(y-1)^2  = 400

Divide both sides of the equation by 400


(25(x+3)^2)/400 + (4(y-1)^2)/400  = 400/400              Note also that, 16*25 = 400.


((x+3)^2)/16 + ((y-1)^2)/100  = 1

((x+3)^2)/(5^2) + ((y-1)^2)/(10^2)  = 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+3) = (x - h) , h = -3.

Comparing   with above   (y-1) = (y - k) , k = 1.

Therefore center = (h,k) = (-3, 1).

You can easily draw the ellipse...Cheers.

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