A line L passing through the point $(2,0)$ makes an angle $60^{\circ}$ with the line $2 x-y+3=0$. If L makes…
A line L passing through the point $(2,0)$ makes an angle $60^{\circ}$ with the line $2 x-y+3=0$. If L makes an acute angle with the positive X -axis in the anticlockwise direction. then the Y -intercept of the line L is
$\frac{10 \sqrt{3}-16}{11}$
$\frac{3 \sqrt{2}}{\sqrt{7}}$
$\frac{16-10 \sqrt{3}}{11}$
$2$
Solution
Equation of line passing through the point $(2,0)$ and makes angle $60^{\circ}$ with $2 x-y+3=0$ is
$y-y_1=\frac{m \pm \tan \alpha}{1 \mp m \tan \alpha}\left(x-x_1\right) \Rightarrow y=\frac{2 \pm \sqrt{3}}{1 \mp 2 \sqrt{3}}(x-2)$
As line makes acute angle with positive $x$-axis
$\Rightarrow y=\frac{2-\sqrt{3}}{1+2 \sqrt{3}}(x-2) \Rightarrow y=\frac{8-5 \sqrt{3}}{-11}(x-2)$
$\Rightarrow y$-intercept $=\frac{16-10 \sqrt{3}}{11}$