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Latus Rectum of Ellipse

If m1 and m2 ...

Question

If $m_{1}$ and $m_{2}$ are the slopes of the tangents drawn from the point $P (6, - 2)$ to the ellipse $4 x^{2} + 9 y^{2} = 36,$ then $m_{1}^{2} + m_{2}^{2}$ is equal to

\frac{16}{9}

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\frac{32}{27}

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\frac{64}{81}

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\frac{32}{9}

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Solution

The correct option is C $\frac{64}{81}$
Given Ellipse: $4 x^{2} + 9 y^{2} = 36$
$\Rightarrow \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
Let equation of tangent is $y = m x + c$
Its passes through $P (6, - 2) .$
$\Rightarrow y = m x - (6 m + 2)$

Condition for the line $y = m x + c$ to be a tangent is
$c^{2} = a^{2} m^{2} + b^{2}$
$\Rightarrow (6 m + 2)^{2} = 9 m^{2} + 4$
$\Rightarrow 36 m^{2} + 24 m + 4 = 9 m^{2} + 4$
$\Rightarrow 27 m^{2} + 24 m = 0$
$\Rightarrow 9 m^{2} + 8 m = 0 \to roots will be m_{1}, m_{2}$
So, $m_{1} = 0, m_{2} = - \frac{8}{9}$
$∴ m_{1}^{2} + m_{2}^{2} = \frac{64}{81}$

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