First slide

Definition of a circle

Question

The range of values of $λ, λ > 0$ such that the angle $θ$ between the pair of tangents drawn from $(λ, 0)$ to the circle $x^{2} + y^{2} = 4$ lies in $(π / 2, 2 π / 3)$ is

Moderate

A

$(4 / \sqrt{3}, 2 \sqrt{2})$
B

$(0, \sqrt{2})$
C

$(1, 2)$
D

none of these

Solution

we have $\frac{π}{2} < θ < \frac{2 π}{3}, i.e., \frac{π}{4} < \frac{θ}{2} < \frac{π}{3}$
or $\frac{1}{\sqrt{2}} < \sin \frac{θ}{2} < \frac{\sqrt{3}}{2}$
But $\sin \frac{θ}{2} = \frac{2}{λ} or \frac{1}{\sqrt{2}} < \frac{2}{λ} < \frac{\sqrt{3}}{2}$
or $\frac{4}{\sqrt{3}} < λ < 2 \sqrt{2}$

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