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Tangent To a Parabola

If y+3= m 1...

Question

If $y + 3 = m_{1} (x + 2)$ and $y + 3 = m_{2} (x + 2)$ are two tangents to the parabola $y^{2} = 8 x$ , 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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None of these

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Solution

The correct option is B $m_{1} m_{2} = - 1$
Equation of tangent to parabola $y^{2} = 4 a x$ is given by,
$y = m x + a / m$ , here $a = 2$
$\Rightarrow y = m x + 2 / m$
Now it passes through $(- 2, - 3)$
$\Rightarrow - 3 = - 2 m + 2 / m$
$\Rightarrow 2 m^{2} - 3 m - 2 = 0$
$m_{1}, m_{2}$ are the roots of above quadratic
$∴ m_{1} + m_{2} = 3 / 2, m_{1} m_{2} = - 2 / 2 = - 1$

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