First slide

Definition of a circle

Question

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

Moderate

A

$\sqrt{P Q . R S}$
B

$\frac{P Q + R S}{2}$
C

$\frac{2 P Q . R S}{P Q + R S}$
D

$\sqrt{\frac{P Q^{2} + R S^{2}}{2}}$

Solution

$∠ P X R = 90 °$
$So, ∠ X P R = θ$
$\Rightarrow ∠ X R P = 90^{°} - θ$
$∴ \tan θ = \frac{S R}{P R} = \frac{S R}{2 r} - - - - - - - - (1)$
$a n d \tan (90 ° - θ) = \frac{P Q}{P R} = \frac{P Q}{2 r}$
$\Rightarrow \cot θ = \frac{P Q}{2 r} - - - - - - - - - (2)$
$\Rightarrow 1 = \frac{P Q R S}{(2 r)^{2}} (b y (1) a n d (2))$
$\Rightarrow 2 r = \sqrt{P Q \cdot R S}$

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