The equation ${a x}^{2} + {b y}^{2} + 2 h x y + 2 g x + 2 f y + c = 0$ represents a circle only if

a = b, h = 0

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a = b \neq 0, h = 0

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a = b \neq 0, h = 0, g^{2} + f^{2} - c > 0

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a = b \neq 0, h = 0, g^{2} + f^{2} - a c > 0

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Solution

The correct option is D $a = b \neq 0, h = 0, g^{2} + f^{2} - c > 0$
$a x^{2} + b y^{2} + 2 h x y + 2 g x + 2 f y + c = 0$
$(x - h)^{2} + (y - k)^{2} = r^{2}$
For a circle $a = b \neq 0$
$h = 0$ , $g^{2} + f^{2} - c > 0$ .

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Similar questions

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

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

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

a = b

h = 0

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

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

\frac{a - b}{h} = \frac{a^{'} - b^{'}}{h^{'}}

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