Question
Consider the circles, $x^{2} + y^{2} = 25$ and $x^{2} + y^{2} = 9$. From the point $A(0,5)$ two segments are drawn touching the inner circle at the points $B$ and $C$ while intersecting the outer circle at the points $D$ and $E$. If ' $O$ ' is the centre of both the circles then the length of the segment $OF$ that is perpendicular to $DE$, is:
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Solution
[Key Concept: Two concentric circles with segments from $A(0,5)$: use power of the point / chord length.]
$sin\theta = \frac{3}{5}$
Slope of $AB,AE$ are $tan\left( \frac{\pi}{2} - \theta \right),tan\left( \frac{\pi}{2} + \theta \right) = \frac{4}{3}, - \frac{4}{3} $Equation of $OB$ is $y = - \frac{3}{4}x$
$$\Rightarrow 4y + 3x = 0 $$$D$ is image of $A$ w.r.t. $OB$
$${\frac{x - 0}{3} = \frac{y - 5}{4} = - 2\frac{4(5)}{25} = - \frac{8}{5} }{D \equiv \left( \frac{- 24}{5},\frac{- 7}{5} \right) \Rightarrow OF = \frac{7}{5} }$$
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