Question

Let $B C$ be the chord of contact of the tangents drawn from a point A to the circle $x^{2}+y^{2}=1$. $P$ is any point on the arc $BC$. Let $PL, PM$ and $PN$ be the lengths of perpendiculars from $P$ on $AB , BC$ and $CA$ respectively, then $PL, PM$ and $PN$ are :

• A. In A.P.
• B. In G.P.
• C. In H.P.
• D. Equal

Answer 1

Answer: (B)

Let the coordinates of $B$ and $C$ be $(\cos \alpha, \sin \alpha)$ and $(\cos \beta, \sin \beta)$ and $(\cos \beta, \sin \beta)$ respectively.
Equations of the tangents at $B$ and $C$ are
$x \cos \alpha+y \sin \alpha=1$ .... (i)
and $x \cos \beta+y \sin \beta=1$ .... (ii)
Equation of the line $BC$ is
$x \cos \left(\frac{\alpha+\beta}{2}\right)+y \sin \left(\frac{\alpha+\beta}{2}\right)=\cos \frac{\alpha+\beta}{2}$ ..... (iii)

Let $P$ be the point $(\cos \theta, \sin \theta)$, then