Question

$\alpha ,\beta ,\gamma$ are the parametric angles of three points A, B, C respectively on the circle $x^2+y^2=1$ and P(-1,0). If the lengths of the chords PA, PB, PC are in G.P. then $cos\left(\frac{\alpha }{2}\right),cos\left(\frac{\beta }{2}\right),cos\left(\frac{\gamma }{2}\right)$ are in

• A. A.P
• B. G.P
• C. H.P
• D. A.G.P

Answer 1

The correct option is B G.P

$A(cos\alpha ,sin\alpha ),A(cos\beta ,sin\beta ),A(cos\gamma ,sin\gamma ),P{A}^2=(cos\alpha +1{)}^2+(sin\alpha -0{)}^2=co{s}^2\alpha +1+2cos\alpha +si{n}^2\alpha =2+2cos\alpha =2(1+cos\alpha )=4co{s}^2\left(\frac{\alpha }{2}\right)\Rightarrow SimilarlyPB=2cos\left(\frac{\beta }{2}\right),PC=2cos\left(\frac{\gamma }{2}\right)PA,PB,PCareinG.P\Rightarrow 2cos\left(\frac{\alpha }{2}\right),2cos\left(\frac{\beta }{2}\right),2cos\left(\frac{\gamma }{2}\right)areinG.P\Rightarrow cos\left(\frac{\alpha }{2}\right),cos\left(\frac{\beta }{2}\right),cos\left(\frac{\gamma }{2}\right)areinG.P.$