Question

A chord $ax+y=1$ subtends ${90}^{\circ }$ at the center of the circle $x^2+y^2=\frac{3}{2}.$ Then the value of $a$ is

• A. $\pm \frac{1}{\sqrt{2}}$
• B. $\pm \frac{1}{\sqrt{3}}$
• C. $\pm \sqrt{2}$
• D. $\pm \sqrt{3}$

Answer 1

The correct option is B $\pm \frac{1}{\sqrt{3}}$


By homoginization,
$x^2+y^2=\frac{3}{2}(ax+y{)}^2\Rightarrow 2x^2+2y^2=3(ax+y{)}^2\Rightarrow (3a^2-2)x^2+y^2+6axy=0$

This pair of straight lines subtends ${90}^{\circ }$ at the origin.
$\Rightarrow \text{Coefficient of}{x}^2+\text{Coefficient of}{y}^2=0\Rightarrow 3a^2-1=0\Rightarrow a=\pm \frac{1}{\sqrt{3}}$