General form of a conic is $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ , if $h^{2} = a b$ , then the conic is

an ellipse

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a hyperbola

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a parabola

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None of the above

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Solution

The correct option is C a parabola
A pair of straight line if $Δ = 0$ , where $Δ = a b c + 2 f g h - a f^{2} - b g^{2} + c = 0$ .
or $Δ = ⎡ ⎢ ⎣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ⎤ ⎥ ⎦$
For $a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} \neq 0$ and $h^{2} = a b$ , conic is parabola.
So, option C is correct

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Similar questions

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

h^{2} > a b

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

h^{2} < a b

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0,

a = b = h = 1

f \neq g,

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

(\frac{b g - h f}{h^{2} - a b}, \frac{a f - g h}{h^{2} - a b})

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

(\sqrt{(\frac{f^{2} - b c}{h^{2} - a b})}, \sqrt{(\frac{g^{2} - a c}{h^{2} - a b})})

\sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}} = 1

a, b \neq 0

A x^{2} + 2 H x y + B y^{2} + 2 G x + 2 F y + c = 0

Δ \neq 0

H^{2} = A B

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