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Slope Form a Line

If the lines ...

Question

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

A

m_{1} + m_{2} = 0

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B

m_{1} m_{2} = 1

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C

m_{1} m_{2} = - 1

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D

m_{1} + m_{2} = 1

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Solution

The correct option is A $m_{1} m_{2} = - 1$
Condition for tangency is $c = \frac{a}{m}$
Hence
$(a m_{1} - b) m_{1} = a$
$(a m_{2} - b) m_{2} = a$
On comparing we get
$a {m_{1}}^{2} - b m_{1} = a {m_{2}}^{2} - b m_{2}$
$a (m_{1}^{2} - m_{2}^{2}) = b (m_{1} - m_{2})$

$a (m_{1} - m_{2}) (m_{1} + m_{2}) = b (m_{1} - m_{2})$

$m_{1} + m_{2} = \frac{b}{a}$
Now $(m_{1} - \frac{b}{a}) m_{1} = 1$
$m_{1} m_{2} = - 1$

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