Question

Coordinates of the foot of the perpendicular drawn from $(0,0)$ to the line joining $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$ are:
A) $\left[ {\dfrac{a}{2}(\cos \alpha - \cos \beta ),\dfrac{a}{2}(\sin \alpha - \sin \beta )} \right]$
B) $\left( {\cos \dfrac{{\alpha + \beta }}{2},\sin \dfrac{{\alpha + \beta }}{2}} \right)$
C) $\left[ {\dfrac{a}{2}(\cos \alpha + \cos \beta ),\dfrac{a}{2}(\sin \alpha + \sin \beta )} \right]$
D) None of these.

Answer 1

Hint: Here, we will be using the concept of direction cosines and direction ratios. If a line passing through origin makes angles $\alpha $, $\beta $ and $\gamma $ with the $x$,$y$ and $z$ axis, then the cosine of these angles named as $\cos \alpha $,$\cos \beta $ and $\cos \gamma $ are called the direction cosines of the line. The direction ratios are any three values that are proportional to a line’s direction cosines.

Complete step by step Solution:
Given, points $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$ join to form a line.
We can write
${(a\cos \alpha )^2} + {(a\sin \alpha )^2} = {a^2}({\cos ^2}\alpha + {\sin ^2}\alpha )$
$\therefore {(a\cos \alpha )^2} + {(a\sin \alpha )^2} = {a^2}$
Also
${(a\cos \beta )^2} + {(a\sin \beta )^2} = {a^2}({\cos ^2}\beta + {\sin ^2}\beta )$
$\therefore {(a\cos \beta )^2} + {(a\sin \beta )^2} = {a^2}$
From both these equations, we can say that the given points are lying on a circle with a radius $a$
${x^2} + {y^2} = {a^2}$
So, if we draw a perpendicular from the origin, then it will bisect the chord joining the given points at the midpoint.
Hence, we can say that the coordinates $(h,k)$ of the foot of perpendicular will be the midpoint of the given two points.
$(h,k) = \left[ {\dfrac{a}{2}(\cos \alpha + \cos \beta ),\dfrac{a}{2}(\sin \alpha + \sin \beta )} \right]$

Therefore, the correct option is (C).

Note: Direction ratios are also referred to as direction numbers. If $l$ , $m$ and $n$ are direction cosines of the line $ax + by + c = 0$ then, $a = \lambda l$ , $b = \lambda m$ and $c = \lambda n$ for any non-zero $\lambda \in \mathbb{R}$ . For any line, there can be infinitely many sets of direction ratios. $l = \pm \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$ , $m = \pm \dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$ and $n = \pm \dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$ .