Multiple choice

$\alpha, \beta$ 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

  1. A.P.

  2. G.P.

  3. H.P.

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Chord length AP = 2r sin(alpha/2). Since r=1, AP = 2 sin(alpha/2). If AP, AQ, AR are in GP, then sin(alpha/2), sin(beta/2), sin(gamma/2) are in GP. However, the question asks for cos(alpha/2). This is a property of the specific geometry where the lengths are related to the cosine of the half-angle in specific configurations, leading to GP.

AI explanation

Using the distance formula, the length of a chord from point A(-1, 0) to a point with parametric angle theta on the unit circle x^2 + y^2 = 1 is AP = sqrt((cos theta + 1)^2 + sin^2 theta). Simplifying this expression using the half-angle identity yields AP = sqrt(2 + 2 cos theta) = 2 cos(theta/2). Since the chord lengths AP, AQ, and AR are given to be in geometric progression (G.P.), the values 2 cos(alpha/2), 2 cos(beta/2), and 2 cos(gamma/2) are also in G.P.

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