4x^2+25y^2+50y-75=0

Question

complete the square.

Answer 1

The equation 4x ² +25y ² + 50 y - 75 = 0

To do this we complete the squre on the y  terms and move the other terms to right.

25y ² + 50y  = 75 - 4x ²

Here y ² coefficient is 25, for perfect square make y ² coefficient 1 by divide each side by 25.

y ²+ 2y = 3 - (4/25)x ²

To change the expression into a perfect square trinomial add (half the y coefficient)² to each side of the expression.

y ² + 2y  + 1 = 3 - (4/25)x ² + 1

y ²+ 2y  + 1 = 4 - (4/25)x ²

 (4x ²/25) + (y + 1) ²  = 4

x ² / 25 + (y + 1)²/4 = 1

(x - 0) ²/5² + (y -(-1))²/4 = 1

Compare it to standard form of  horizontal  ellipse (x - h )²/a ²  + (y - k )²/b ² = 1 where a > b .

Center (h, k ) , a  is semi major axis and b  is semi minor axis.

(h, k ) = (0 , -1) a = 5 and b = -1.