First slide

Definition of a circle

Question

$α, β$ and y are the 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 $A P, A Q$ and $A R$ are in $G P,$ then $\cos α / 2, \cos β / 2$ and $\cos γ / 2$ are in

Moderate

A

AP
B

GP
C

HP
D

none of these

Solution

Let $P (\cos α, \sin α), Q (\cos β, \sin β)$ and $R (\cos γ, \sin γ)$ be three specified points on the given circle. Then,
$\begin{array}{l} A P = \sqrt{(- 1 - \cos α)^{2} + (0 - \sin α)^{2}} \\ \Rightarrow A P = \sqrt{2 + 2 \cos α} = \sqrt{4 \cos^{2} α / 2} = 2 \cos α / 2 \end{array}$
Similarly, we have
$A Q = 2 \cos β / 2$ and $A R = 2 \cos γ / 2$
Now,
$A P, A Q, A R$ are in GP
$\Rightarrow 2 \cos α / 2, 2 \cos β / 2, 2 \cos γ / 2$ are in GP
$\Rightarrow \cos α / 2, \cos β / 2, \cos γ / 2$ are in GP.

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