First slide

Pair of straight lines

Question

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

Moderate

A

$m^{2} = 1$
B

$m^{2} = 3$
C

$m^{2} = 7$
D

$2 m^{2} = 1$

Solution

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
$\begin{array}{l} x^{2} + y^{2} = {(\frac{y - m x}{2})}^{2} \\ \Rightarrow x^{2} (4 - m^{2}) + 2 m x y + 3 y^{2} = 0 \end{array}$
Since these lines are at right angles
$4 - m^{2} + 3 = 0 \Rightarrow m^{2} = 7$

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