# Math Help - Hyperbola and circle

1. ## Hyperbola and circle

Hyperbola $x^2-y^2+ax+by=0$ and circle $x^2+y^2=a^2+b^2$ intersect at four points. Show that three of them are vertices of an equilateral triangle.

2. ## Hyperbola and circle

Hello atreyyu
Originally Posted by atreyyu
Hyperbola $x^2-y^2+ax+by=0$ and circle $x^2+y^2=a^2+b^2$ intersect at four points. Show that three of them are vertices of an equilateral triangle.
Let $x=r\cos \theta$ and $y=r\sin\theta$, where $r^2=a^2+b^2$.

Then $(x,y)$ lies on the circle for all $\theta$.

Circle intersects hyperbola where:

$r^2\cos^2\theta - r^2\sin^2\theta = -ar\cos\theta-br\sin\theta$

$\implies \cos^2\theta - \sin^2\theta = -(\frac{a}{r}\cos\theta+\frac{b}{r}\sin\theta)$

Now express the LHS as $\cos 2\theta$, and the RHS as $-\cos(\theta -\alpha)$, where $\tan\alpha=...$

Then show that there are three values of $\theta$ that differ by $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$.

Can you do it now?