First slide

Definition of a circle

Question

The length of the common chord of the circles $x^{2} + y^{2} + 4 x + 1 = 0$ and $x^{2} + y^{2} + 4 y - 1 = 0,$ is

Moderate

A

$\sqrt{15 / 2}$
B

$\sqrt{15}$
C

$2 \sqrt{15}$
D

none of these

Solution

Given circles intersect orthogonally. So, the length of their common chord is
$l = \frac{2 r_{1} r_{2}}{\sqrt{r_{1}^{2} + r_{2}^{2}}},$
where $r_{1}$ and $r_{2}$ are the radii of the given circles.
Here, $r_{1} = \sqrt{5}$ and $r_{2} = \sqrt{3} .$
$∴ l = \frac{2 \sqrt{15}}{\sqrt{5 + 3}} = \sqrt{\frac{15}{2}} . .$

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