First slide

Definition of a circle

Question

If the length of the chord of the circle $x^{2} + y^{2} = r^{2}$ along the line $y - 2 x = 3$ is $r,$ then $r^{2}$ is equal to

Moderate

A

$\frac{9}{5}$
B

12
C

$\frac{12}{5}$
D

$\frac{24}{5}$

Solution

The length of the chord of the circle
$x^{2} + y^{2} = r^{2}$ along the line $y = m x + c$ is given by
$2 \sqrt{\frac{r^{2} (1 + m^{2})}{1 + m^{2}} - c^{2}}$
Here, $m = 2$ and $c = 3 .$ Therefore,
$\begin{array}{l} r = 2 \sqrt{\frac{r^{2} (1 + 4) - 9}{1 + 4}} \\ \Rightarrow r = 2 \sqrt{\frac{5 r^{2} - 9}{5}} \Rightarrow 5 r^{2} = 20 r^{2} - 36 \Rightarrow r^{2} = \frac{36}{15} = \frac{12}{5} \end{array}$

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