Question

Obtain the equation of the line passing through $(2,6)$, given that the length of the perpendicular segment from the origin to the line is $2$.

Answer 1

Equation of line whose perpendicular distance from origin is $2$ - $x\cos \theta +y\sin \theta =2$

$\therefore x=2$ ...(i)

$\pm \frac{3x}{5}-\frac{4y}{5}=2$ ....(ii)

$\therefore 2\cos \theta +6\sin \theta =1$

$\Rightarrow \cos \theta +3\sin \theta =1$

$\Rightarrow 3\sin \theta =1-\cos \theta$

$\Rightarrow 3\sqrt{1-{\cos }^2\theta }=1-\cos \theta$

$\Rightarrow 9(1-{\cos }^2\theta )=1-2\cos \theta +{\cos }^2\theta$

$\Rightarrow 10{\cos }^2\theta -2\cos \theta -8=0$

$\Rightarrow 5{\cos }^2\theta -\cos \theta -4=0$

$\Rightarrow \cos \theta =1,-\frac{4}{5}$

$\Rightarrow \sin \theta =0,\pm \frac{3}{5}$