First slide

Analysis of two circles in circles

Question

The angle of intersection of the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} = 2 x + 2 y$ , is

Moderate

A

$π / 2$
B

$π / 3$
C

$π / 6$
D

$π / 4$

Solution

We know that the angle of intersection of two circles of radii $r_{1}$ and $r_{2}$ is given by
$\cos θ = \frac{r_{1}^{2} + r_{2}^{2} - d^{2}}{2 r_{1} r_{2}}$ ,
where d is the distance between their centres
Here, $r_{1} = 2, r_{2} = \sqrt{2}$ and $d = \sqrt{2}$ .
$∴ \cos θ = \frac{4 + 2 - 2}{2 \times 2 \times \sqrt{2}} = \frac{1}{\sqrt{2}} \Rightarrow θ = \frac{π}{4}$

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