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.
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 . For the final result you'll want to take a=4 and b=3.
A line through the point , with slope , has equation . The line meets the ellipse at points where That equation is a quadratic in x, namely 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 , namely .
That equation has two roots, say and , which are the slopes of the two tangents to the ellipse from the point . You want them to be perpendicular, in other words . But the product of the roots of a quadratic equation is the constant term divided by the coefficient of . That gives you the condition , which tells you that lies on the circle .