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.
