The curve $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ intercepts on the line lx+ my = 1, a length which subtends a right angle at origin then $\frac{l g + m f + 1}{l^{2} + m^{2}} =$

- c

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- c/2

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c/2

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Solution

The correct option is C - c/2
Let the intercepts be $A$ and $B$ and $O$ be the origin.
Homogenisation gives $O A \times O B$ equation
$l x + m y = 1$
$\Rightarrow x^{2} + y^{2} + (2 g x + 2 f y) (l x + m y) + c (l x + m y)^{2} = 0$
$\Rightarrow x^{2} + y^{2} + 2 g l x^{2} + 2 g m x y + 2 f l x y + 2 m f y^{2} + c (l^{2} x^{2} + m^{2} y^{2} + 2 m l x y) = 0$
$\Rightarrow x^{2} (c l^{2} + 1 + 2 g l) + y^{2} (c m^{2} + 2 m f + 1) + x y (2 m l + 2 g m + 2 f l) = 0$
$m_{1} m_{2} = \frac{a}{b} = - 1$
$\Rightarrow c l^{2} + 2 g l + 1 = - (c m^{2} + 2 m f + 1)$
$\Rightarrow c (l^{2} + m^{2}) = - (2 g l + 2 m f + 2)$
$\Rightarrow c (l^{2} + m^{2}) = - 2 (g l + m f + 1)$
$\Rightarrow \frac{l g + m f + 1}{l^{2} + m^{2}} = \frac{- c}{2}$

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