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Math Help - Hyperbola and circle

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    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.
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    Hyperbola and circle

    Hello atreyyu
    Quote Originally Posted by atreyyu View Post
    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?

    Grandad
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