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Implicit Differentiation

If m1, m2 b...

Question

If $m_{1}, m_{2}$ be the slopes of the two tangents drawn from $(1, 2)$ to the ellipse $2 x^{2} + 3 y^{2} = 6$ then $m_{1} + m_{2} =$

3

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Solution

The correct option is B $- 2$
Ellipse equation is $2 x^{2} + 3 y^{2} = 6$

Dividing by 6. We get,
$\frac{x^{2}}{3} + \frac{y^{2}}{2} = 1$

Equation of line through $(1, 2)$ is
$(y - 2) = m (x - 1)$
$\Rightarrow y = m x + (2 - m)$
But condition for a line $y = m x + c$ to be a tangent is

$c^{2} = a^{2} m^{2} + b^{2} .$
$\Rightarrow (2 - m)^{2} = 3 m^{2} + 2$
$\Rightarrow 2 m^{2} + 4 m - 2 = 0$
$∴ m_{1} + m_{2} = \frac{- b}{a} = \frac{- 4}{2} = - 2 \dots (s u m o f r o o t s)$

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