First slide

Analysis of two circles in circles

Question

The circles $x^{2} + y^{2} + x + y = 0$ and $x^{2} + y^{2} + x - y = 0$ intersect at an angle of

Moderate

Solution

The angle 8 of intersection of two cirlces is given by
$\cos θ = \frac{r_{1}^{2} + r_{2}^{2} - C_{1} C_{2}^{2}}{2 r_{1} r_{2}}$
where $r_{1}, r_{2}$ are radii of two circles and $C_{1} C_{2}$ is the distance between their centres.
Here, $r_{1} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{1}{2}} = r_{2}$ and $C_{1} C_{2} = 1$
$∴ \cos θ = 0 \Rightarrow θ = \frac{π}{2}$
$A L I T E R$ For the two circles, we find that $2 (g_{1} g_{2} + f_{1} f_{2}) = c_{1} + c_{2}$
hold. So, the angle of intersection is a right angle.

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