Question

The pair of straight lines joining the origin to the points of intersection of the line $y=2\sqrt{2}x+c$ and the circle $x^2+y^2=2$ are at right angles, if

• A. $c^2-4=0$
• B. $c^2-8=0$
• C. $c^2-9=0$
• D. $c^2-10=0$

Answer 1

The correct option is C

$c^2-9=0$

Pair of straight lines joining the origin to the points of intersection of the line $y=2\sqrt{2}x+c$ and the circle $x^2+y^2=2$ are

$\Rightarrow x^2+y^2+(-2){\left(\frac{2\sqrt{2}x-y}{-c}\right)}^2=0$

$\Rightarrow x^2+y^2-\frac{2}{c^2}(8x^2+y^2-4\sqrt{2}xy)=0$

$\Rightarrow x^2(1-\frac{16}{c^2})+y^2(1-\frac{2}{c^2})+\frac{8\sqrt{2}xy}{c^2}=0$

If these lines are perpendicular, $1-\frac{16}{c^2}+1-\frac{2}{c^2}=0$

$\Rightarrow \frac{2c^2-18}{c^2}=0\Rightarrow c^2-9=0.$