Fun of Thinking — Math Problems from Around the World

Tangents from the Origin to a Circle on an Ellipse

Problem

As shown in the figure, in the coordinate plane $xOy$ , let $M(x_0, y_0)$ be a point on the ellipse

C: \dfrac{x^2}{4} + y^2 = 1,

whose left and right foci are $F_1$ and $F_2$ . From the origin $O$ , two tangent lines are drawn to the circle

M: (x - x_0)^2 + (y - y_0)^2 = r^2 \qquad (0 < r < 1),

meeting the ellipse $C$ at points $P$ and $Q$ , respectively. Let $k_1$ and $k_2$ denote the slopes of lines $OP$ and $OQ$ .

1. Suppose lines $MF_1$ and $MF_2$ meet the circle $M$ at points $A$ and $B$ , respectively. When $|AF_1| - |BF_2| = 2r$ , find the equation of the locus of $A$ . 2. If $k_1 k_2$ is a constant (independent of the position of $M$ ), find the maximum value of $|OP| \cdot |OQ|$ .

Answer

Solution

Difficulty	7/10
Topics	Extrema, Parametrization, conic sections, analytic geometry, Vieta's Formulas

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