What type of conic is created by 4x-5y plus x squared plus 2y squared equals 0?

Answer 1

4x - 5y + x2 + 2y2 = 0; rewrite as

1x2 + 2y2 + 4x - 5y + 0 = 0;

Since the coefficient of x and y are different, and their product is greater than zero, the given equation is the equation of an ellipse.

add 4 to both sides to complete the square for x, and factor out 2 from y-terms

(x + 2)2 + 2[y2 - (5/2)y] = 4; divide by 2 each term to both sides

(x + 2)2/2 + y2 - (5/2)y = 2; add (5/4)2 to both sides to complete the square for y

(x + 2)2/2 + (y - 5/4)2 = (2*16 + 25)/16

(x + 2)2/2 + (y - 5/4)2 = 57/16; let + 2 = - (-2) and divide by 57/16 to both sides

[x - (-2]2/(57/8) + (y - 5/4)2/(57/16) = 1

This is the equation of an ellipse centered at (-2, 5/4).