Question

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

• A. $\frac{1}{2}r$
• B. $r$
• C. $\frac{1}{2}L$
• D. $\frac{2}{3}L$

Answer 1

The correct option is C $\frac{1}{2}L$

Given the length of tangent L = $\frac{4}{3}$r, where r is the radius
or $r=\frac{3L}{4}$
From the figure $L^2+{\left(\frac{3L}{4}\right)}^2={(x+\frac{3L}{4})}^2$
$L^2+{\left(\frac{3L}{4}\right)}^2=x^2+{\left(\frac{3L}{4}\right)}^2+2x\left(\frac{3L}{4}\right)$
$x^2+2\left(\frac{3L}{4}\right)x-L^2=0$
$2x^2+3Lx-2L^2=0$
or $(x+2L)(2x-L)=0$
$\Rightarrow x=-2Lor\frac{L}{2}$
We reject $x=-2L$. Hence $x=$ $\frac{L}{2}$