Question

$\alpha ,\beta$ and $\gamma$ are parametric angles of three points $P,Q$ and $R$ respectively, on the circle $x^2+y^2=1$ and $A$ is the point $(-1,0)$. If the lengths of the chords $AP,AQ$ and $AR$ are in G.P., then $\cos \alpha /2,\cos \beta /2$ and $\cos \gamma ./2$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is B G.P.

Coordinates of $P.Q,R$ are $(\cos \alpha ,\sin \alpha ),(\cos \beta ,\sin \beta )$ and $(\cos \gamma ,\sin \gamma )$ respectively and $A\equiv (-1,0)$

$\therefore AP=\sqrt{{(1+\cos \alpha )}^2+{\sin }^2\alpha }=\frac{2\cos \alpha }{2}$

$AQ=\sqrt{{(1+\cos \beta )}^2+{\sin }^2\beta }=\frac{2\cos \beta }{2}$

$AR=\sqrt{{(1+\cos \gamma )}^2+{\sin }^2\gamma }=\frac{2\cos \gamma }{2}$

$\because AP,AQ,AR$ are in $G.P.$ then $\frac{\cos \alpha }{2},\frac{\cos \beta }{2},\frac{\cos \gamma }{2}$ are also in $G.P.$