A Cassini Oval

★ 7/10Extrema Substitution conic sections analytic geometry

Problem

A Cassini oval is the locus of points whose distances to two fixed points have constant product. In the coordinate plane, let $M(-2, 0)$ and $N(2, 0)$ , and let $P$ satisfy

|PM| \cdot |PN| = 5.

Determine, with proof, which of the following are true:

1. the $x$ -coordinate of $P$ ranges over $\left[-\sqrt{5}, \sqrt{5}\right]$ ; 2. $|OP|$ ranges over $[1, 3]$ ; 3. the maximum area of triangle $PMN$ is $\dfrac{5}{2}$ ; 4. $|PM| + |PN|$ ranges over $\left[2\sqrt{5}, 5\right]$ .

Answer

Solution

Difficulty	7/10
Topics	Extrema, Substitution, conic sections, analytic geometry

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