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Thread: Ellipses and Circles

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    Ellipses and Circles

    An ellipse is drawn with maximum radius 4 cm and minimum radius 3 cm. A circle is drawn, as shown in the figure, such that any two tangents of the ellipse that meet on the circle make 90⁰ with each other. Find the radius of the circle.
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    Quote Originally Posted by rickrishav View Post
    An ellipse is drawn with maximum radius 4 cm and minimum radius 3 cm. A circle is drawn, as shown in the figure, such that any two tangents of the ellipse that meet on the circle make 90⁰ with each other. Find the radius of the circle.
    Here is an outline of one way to approach this problem. I'll work with the ellipse $\displaystyle \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} = 1$. For the final result you'll want to take a=4 and b=3.

    A line through the point $\displaystyle (x_0,y_0)$, with slope $\displaystyle \lambda$, has equation $\displaystyle y-y_0 = \lambda(x-x_0)$. The line meets the ellipse at points where $\displaystyle \dfrac{x^2}{a^2}+\dfrac{(y_0 + \lambda(x-x_0))^2}{b^2} = 1.$ That equation is a quadratic in x, namely $\displaystyle (b^2+\lambda^2a^2)x^2 + 2\lambda a^2(y_0 -\lambda x_0)x + a^2((y-\lambda x)^2 - b^2) = 0.$ The condition for the line to be a tangent to the ellipse is that the equation for x should have equal roots, in other words "b^2 – 4ac=0". Write down that condition and simplify it, to get a quadratic equation in $\displaystyle \lambda$, namely $\displaystyle (x_0^2-a^2)\lambda^2 -2x_0y_0\lambda + y_0^2-b^2 = 0$.

    That equation has two roots, say $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$, which are the slopes of the two tangents to the ellipse from the point $\displaystyle (x_0,y_0)$. You want them to be perpendicular, in other words $\displaystyle \lambda_1\lambda_2 = -1$. But the product of the roots of a quadratic equation is the constant term divided by the coefficient of $\displaystyle \lambda^2$. That gives you the condition $\displaystyle \dfrac{y_0^2-b^2}{x_0^2-a^2} = -1$, which tells you that $\displaystyle (x_0,y_0)$ lies on the circle $\displaystyle x^2+y^2 = a^2+b^2$.
    Last edited by Opalg; Dec 9th 2010 at 12:25 AM.
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