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Modulus Function

If y+3=m1x+...

Question

If $y + 3 = m_{1} (x + 2)$ and $y + 3 = m_{2} (x + 2)$ are two tangents to the parabola $y^{2} = 8 x$ , prove that $m_{1} m_{2} = - 1$ .

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Solution

Condition for tangents is c = a/m
so for first case $2 m_{1} - 3 = \frac{2}{m_{1}}$
and $2 m_{2} - 3 = \frac{2}{m_{2}}$
subtracting, $2 m_{1} - 3 - 2 m_{2} + 3 = \frac{2}{m_{1}} - \frac{2}{m_{2}}$
$2 (m_{1} - m_{2}) = 2 (\frac{1}{m_{1}} - \frac{1}{m_{2}})$
$\Rightarrow m_{1} - m_{2} = \frac{m_{2} - m_{1}}{m_{1} m_{2}}$
$\Rightarrow m_{1} - m_{2} = - \frac{m_{1} - m_{2}}{m_{1} m_{2}}$
$∴ m_{1} m_{2} = - 1$ Hence proved

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