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Angle Subtended by an Arc of a Circle on the Circle and at the Center

Two concentri...

Question

Two concentric circles with centre $C$ have radii $r_{1}$ and $r_{2}$ such that $(r_{2} - r_{1}) > 0$ . $C A$ and $C B$ are the common lined radii of the circles. If tangent at $A$ is drawn to meet the bigger circle in the point $D$ . Then the length $B D$ is given by

A

\sqrt{2 r_{1} (r_{2} - r_{1})}

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B

\sqrt{2 r_{2} (r_{2} - r_{1})}

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C

\sqrt{2 r_{1} (r_{2} + r_{1})}

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D

\sqrt{2 r_{2} (r_{2} + r_{1})}

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Solution

The correct option is B $\sqrt{2 r_{2} (r_{2} - r_{1})}$
$A B = r_{2} - r_{1}$
From $△ D C A, ∠ D A C = 90^{o}$
$\Rightarrow D A = \sqrt{r_{2}^{2} - r_{1}^{2}}$
From $△ D A B, ∠ D A B = 90^{o}$
$\Rightarrow B D = \sqrt{r_{2}^{2} - r_{1}^{2} + {(r_{2} - r_{1})}^{2}}$
$\Rightarrow B D = \sqrt{r_{2}^{2} - r_{1}^{2} + r_{2}^{2} + r_{1}^{2} - 2 r_{1} r_{2}} = \sqrt{2 r_{2} (r_{2} - r_{1})}$

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