Question

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

• A. $m = \sqrt{3}$
• B. $m = \pm \sqrt{7}$
• C. $m = \sqrt{1}$
• D. $m = \sqrt{5}$

Answer 1

Answer: (B)

$m = \pm \sqrt{7}$

Answer 2

Given equation of line $y = mx + 2$ or $\frac{y - mx}{2} = 1$ and circle $x^2 + y^2 = 1$ $\therefore$ Equation of the line joining the origin to the intersection of line and circle is $x^2 + y^2 - \left(\frac{y - mx}{2}\right)^{2} = 0$ $\Rightarrow 4\left(x^{2} + y^{2}\right) -\left(y^{2} + m^{2} x^{2} - 2myx\right) = 0$ $\Rightarrow x^{2}\left(4-m^{2}\right) + 3y^{2 } + 2mxy= 0$ Since, lines are at right angles. $\therefore 4 -m^{2 } + 3 = 0 \left( \because a + b = 0\right)$ $\Rightarrow m^{2 } =7$ $\Rightarrow m = \pm\sqrt{7}$