If the pair of lines given by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ intersect on $y -$ axis then,

2 f g h = b g^{2} + c h^{2}

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b g^{2} \neq c h^{2}

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a b c = 2 f g h

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None of these

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Solution

The correct option is B $2 f g h = b g^{2} + c h^{2}$
Put x = 0 in the given equation
=> $b y^{2} + 2 f y + c = 0$
For unique point of intersection.
Discriminant = 0
$f^{2} — b c = 0$
=> $a f^{2} — a b c = 0$
Since $a b c + 2 f g h — a f^{2} — b g^{2} — c h^{2} = 0$
=> $2 f g h — b g^{2} — c h^{2} = 0$
$∴ 2 f g h = b g^{2} + c h^{2}$

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