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If I know the following parameters how to find h co-ordinate of ellipse center

1.Circle : center (0,0), radius = r 2.Ellipse : center (h,k), semi-major axis = a and semi-minor axis = b 3.I know that ellipse is tangent to circle.

If I know all parameters, except h , what is the formula /equation find h ?

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The circle is $x^2+y^2=r^2$, the ellipse $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ Then taking implicit derivatives $y'=-\frac{x}{y}$ for the circle and $y'=-\frac{b^2(x-h)}{a^2(y-k)}$ for the ellipse. You get another equation by equating them $\frac{x}{y}=\frac{b^2(x-h)}{a^2(y-k)}$ Pick your favorite pair to solve simultaneously, but it is a mess. You expect four solutions (some may be complex) and tangency will happen when there is a double root.

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  • $\begingroup$ This of course assumes that the axes of the ellipse are parallel to the coordinate axes; if not, the situation is even messier. $\endgroup$
    – J. M. ain't a mathematician
    Commented May 14, 2011 at 16:17

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