Animated Solution for Mathematics - Conic Sections: If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and $P$ be a point on the ellipse, then $\min (SP\cdot S^{\prime }P)+\max (SP\cdot S^{\prime }P)$ is equal to :

We begin with the equation of our ellipse:

By comparing this to the standard form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, we identify our parameters: $a^2=18$ and $b^2=9$. These values represent the "DNA" of our ellipse, defining its stretch along the $x$-axis and compression along the $y$-axis.

Many students reach for the coordinate distance formula, but that is often inefficient. Instead, we use the focal distance property: for any point $P(x,y)$ on the ellipse, the distances to the foci are $SP=a-ex$ and $S^{\prime }P=a+ex$.

As you move along the ellipse, the $x$-coordinate dictates how these distances fluctuate. When we multiply these two, we obtain the product:

The expression $(a-ex)(a+ex)$ is a classic difference of squares. Expanding this, we get:

This product is a function of $x^2$. Because the $x$-coordinate for any point on the ellipse is constrained to the interval $[-a,a]$, the value of $x^2$ ranges from $0$ to $a^2$.

To maximize the product $a^2-e^2{x}^2$, we subtract the smallest possible value. Since $e^2$ is positive, we set $x^2=0$ (the minor axis):

To minimize the product, we subtract the largest possible value. Setting $x^2=a^2$ (the vertices), we get:

Recall the fundamental relationship for an ellipse: $b^2=a^2(1-e^2)$. Substituting this into our minimum expression, we find:

The bounds of the product are $18$ and $9$. The sum of these two values is: