If the lines $(y - b) = m_{1} (x + a)$ and $(y - b) = m_{2} (x + a)$ are the tangents to the parabola $y^{2} = 4 a x$ , then

m_{1} + m_{2} = 0

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m_{1} m_{2} = 1

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m_{1} m_{2} = - 1

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m_{1} + m_{2} = 1

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Solution

The correct option is C $m_{1} m_{2} = - 1$
By the equation, it is clear that two tangents are drawn from point (-a,b)

Hence the equation of the two tangents can be written as $\Rightarrow$ $\frac{y - b}{x + a} = m$

$\Rightarrow$ $y = m x + a m + b$

For a parabola $y^{2} = 4 a x$ the tangent of form $y = m x + c$

$c = \frac{a}{m}$

Hence we have $\Rightarrow \frac{a}{m} = a m + b$

$\Rightarrow a m^{2} + b m - a = 0$

So $m_{1} . m_{2} = - 1$

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