Question

The length of the perpendicular from the point $\left(a\cos \alpha ,a\sin \alpha \right)$, upon the straight line $y=x\tan \alpha +c$ (where $c>0$ ) is

• A. $c$
• B. $c{\sin }^2\alpha$
• C. $c\cos \alpha$
• D. $cse{c}^2\alpha$

Answer 1

The correct option is C

$c\cos \alpha$

Find the length of perpendicular based on given information

Given straight line, $y=x\tan \alpha +c$

$\Rightarrow$ $x\tan \alpha -y+c=0$ is the equation of the given line

Let $A=\left(x_0,y_0\right)=\left(a\cos \alpha ,a\sin \alpha \right)$

The perpendicular distance of a point from a line is given as

$d=\frac{\left|a{x}_0+b{y}_0+c\right|}{\sqrt{a^2+b^2}}$ where $\left(x_0,y_0\right)$ are co-ordinates of the point and

$ax+by+c=0$is the equation of the line

$\therefore$ $d=\frac{\left|\tan \alpha \times a\cos \alpha -1\times a\sin \alpha +c\right|}{\sqrt{{\tan }^2\alpha +1}}$

$=\frac{\left|a\sin \alpha -a\sin \alpha +c\right|}{\sqrt{se{c}^2\alpha }}$

$\therefore$ $d=\frac{c}{sec\alpha }=c\cos \alpha$

So, the length of the perpendicular from the given point on the given line is $c\cos \alpha$.

Hence, option$\left(C\right)$ is the correct answer.