Multiple choice

The range of values of $\lambda (\lambda > 0)$ such that the angle $\theta $ between the pair of tangents drawn from $(\lambda, 0)$ to the circle $x^2 + y^2 = 4$ lies in $\displaystyle \left ( \frac{\pi}{2} , \frac{2 \pi}{3} \right )$ is

  1. $\displaystyle \left ( \frac{4}{\sqrt{3}} , 2 \sqrt{2} \right )$
  2. $(0, \sqrt{2})$
  3. $(1, 2)$

  4. None of these
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A Correct answer
Explanation

For a circle x^2 + y^2 = 4, the radius is 2. The angle theta between tangents from (lambda, 0) satisfies sin(theta/2) = r/d = 2/lambda. Given pi/2 < theta < 2pi/3, then pi/4 < theta/2 < pi/3. Thus, sin(pi/4) < 2/lambda < sin(pi/3), which is 1/sqrt(2) < 2/lambda < sqrt(3)/2. Solving for lambda gives 4/sqrt(3) < lambda < 2*sqrt(2).

AI explanation

The circle has center at the origin and radius 2, so the length of the tangent from the point lambda comma 0 is the square root of lambda squared minus 4. In the right triangle formed by the tangent, the radius, and the line from the origin to the external point, sin half theta equals radius divided by distance, which is 2 divided by lambda. Since theta lies between pi over 2 and 2pi over 3, half theta lies between pi over 4 and pi over 3. Taking the sine gives 1 over root 2 is less than 2 over lambda is less than root 3 over 2, so lambda is between 4 over root 3 and 2 root 2.

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