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Asymptotes

If the lines ...

Question

If the lines joining the origin to the intersection of the line $y = m x + 2$ and the curve $x^{2} + y^{2} = 1$ are at right angles, then

m^{2} = 1

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m^{2} = 3

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m^{2} = 7

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2 m^{2} = 1

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Solution

The correct option is C $m^{2} = 7$
Joint equation of the lines joining the origin and the point of intersection of the line $y = m x + 2$ and
the curve $x^{2} + y^{2} = 1$ is
$x^{2} + y^{2} = {(\frac{y - m x}{2})}^{2}$
$x^{2} (4 - m^{2}) + 2 m x y + 3 y^{2} = 0$
Since these lines are at right angles
$4 - m^{2} + 3 = 0 \Rightarrow m^{2} = 7$

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