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Distance between Parallel Lines

If y = m1 x...

Question

If $y = m_{1} x + c$ and $y = m_{2} x + c$ are two tangents to the parabola $y^{2} + 4 a (x + a) = 0$ , then

A

m_{1} + m_{2} = 0

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B

1 + m_{1} + m_{2} = 0

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C

m_{1} m_{2} - 1 = 0

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D

1 + m_{1} m_{2} = 0

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Solution

The correct option is D $1 + m_{1} m_{2} = 0$
Given equation of parabola is
$y^{2} + 4 a (x + a) = 0$
$\Rightarrow y^{2} = - 4 a (x + a)$
Vertex of parabola is shifted $a$ units to the right.
Equation of directrix is $x = 0$ i.e. $y -$ axis.
Equation of tangents are
$y = m_{1} x + c$
$y = m_{2} x + c$
Since these tangents intersect at $(0, c)$ which lies on directrix.
Hence tangents are perpendicular.
$\Rightarrow m_{1} m_{2} = - 1$
or, $m_{1} m_{2} + 1 = 0$

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