Determine the type of conic :

Question

y + xy - y^2 - x^2 = 5x + x^2 -y^2?

 

A) Parabola
B) Hyperbola
C) Ellipse
D) not a conic

 

Answer 1

The equation is y + xy - y² - x² = 5x + x² - y² .

y + xy - y² - x² - 5x - x² + y² = 0

y + xy  - 2x² - 5x = 0

y(1 + x)  - x(2x + 5) = 0

y(1 + x)  = x(2x + 5)

Squaring on both sides we get,

y²(1 + x)²  = x²(2x + 5)²

y² / (2x + 5)²  = x² / (1 + x)²

y² / (2x + 5)²  - x² / (1 + x)² = 0

The above equation is not a conic since the right hand side is not 1 and we are unable to make constants a and b.

Therefore the answer is D.