Question

Let $BC$ 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