Multiple choice

The length 'L' of a tangent, drawn from a point 'A' to a circle is $\displaystyle \dfrac{4}{3}$ of the radius $r$. The shortest distance from A to the circle is

  1. $\displaystyle \frac{1}{2}r$
  2. $r$
  3. $\displaystyle \frac{1}{2}L$
  4. $\displaystyle \frac{2}{3}L$
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C Correct answer
Explanation

Let O be the center, P be the point. Tangent length L = 4/3 r. Distance OP = sqrt(r^2 + L^2) = sqrt(r^2 + 16/9 r^2) = sqrt(25/9 r^2) = 5/3 r. Shortest distance to circle = OP - r = 5/3 r - r = 2/3 r. Since L = 4/3 r, then 2/3 r = 1/2 L.

AI explanation

The length of the tangent is given as L = (4/3)r, where r is the radius of the circle, and the distance d from point A to the center of the circle is found using the Pythagorean theorem: d = sqrt(L^2 + r^2). Substituting L gives d = sqrt((16/9)r^2 + r^2) = sqrt(25/9 r^2) = (5/3)r. The shortest distance from point A to the circle is the distance to the center minus the radius, which is (5/3)r - r = (2/3)r. Since L = (4/3)r, we can express the shortest distance in terms of L as (1/2)L.

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