If S and S' 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 $(SP \cdot S'P)$ + max $(SP \cdot S'P)$ is equal to :
Ellipse Equation Parameters
The equation of the ellipse is given as: $ \frac{x^2}{18} + \frac{y^2}{9} = 1 $ From the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, we identify: $a^2 = 18$ $b^2 = 9$
The distance from the center to the foci ($c$) is calculated using $c^2 = a^2 - b^2$. $ c^2 = 18 - 9 = 9 $ So, $c = \sqrt{9} = 3$. The foci $S$ and $S'$ are located at $(\pm c, 0)$, which are $(-3, 0)$ and $(3, 0)$.
Foci Distances Property
For any point $P$ on the ellipse, the sum of the distances from $P$ to the two foci is constant and equal to $2a$. $ SP + S'P = 2a $ Since $a^2 = 18$, $a = \sqrt{18} = 3\sqrt{2}$. Therefore, $SP + S'P = 2(3\sqrt{2}) = 6\sqrt{2}$.
Analyzing Product SP ⋅ S'P
Let $P$ be a point on the ellipse. Using the parametric coordinates $P(a \cos \theta, b \sin \theta)$, the distances from the foci are given by: $SP = a + c \cos \theta$ $S'P = a - c \cos \theta$
The product $SP \cdot S'P$ can be expressed as: $ SP \cdot S'P = (a + c \cos \theta)(a - c \cos \theta) $ $ SP \cdot S'P = a^2 - c^2 \cos^2 \theta $
We know that the value of $\cos^2 \theta$ ranges from $0$ to $1$.
Minimum and Maximum Product Values
To find the maximum value of $SP \cdot S'P$, we minimize the term $c^2 \cos^2 \theta$. This occurs when $\cos^2 \theta = 0$. $ \text{max} (SP \cdot S'P) = a^2 - c^2(0) = a^2 $ To find the minimum value of $SP \cdot S'P$, we maximize the term $c^2 \cos^2 \theta$. This occurs when $\cos^2 \theta = 1$. $ \text{min} (SP \cdot S'P) = a^2 - c^2(1) = a^2 - c^2 $
Final Sum Calculation
Substitute the values of $a^2$ and $c^2$: $a^2 = 18$ $c^2 = 9$ $ \text{max} (SP \cdot S'P) = 18 $ $ \text{min} (SP \cdot S'P) = 18 - 9 = 9 $
The question asks for the sum: min $(SP \cdot S'P)$ + max $(SP \cdot S'P)$. $ \text{Sum} = 9 + 18 = 27 $
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