Key Concepts and Formulas for Ellipse

Before we delve into the solution, let's establish the fundamental properties of an ellipse centered at the origin, which are crucial for solving this problem. For an ellipse with its major axis along the x-axis:

1. Standard Equation: The equation is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b$.

   • $a$: Length of the semi-major axis (half the length of the longest diameter).
   • $b$: Length of the semi-minor axis (half the length of the shortest diameter).

2. Eccentricity ($e$): This parameter quantifies how "elongated" the ellipse is. It's related to $a$ and $b$ by the formula: $b^2 = a^2(1-e^2) \quad \text{or equivalently} \quad e = \sqrt{1 - \frac{b^2}{a^2}}$ For an ellipse, $0 < e < 1$.

3. Foci ($S, S^{\prime}$): These are two fixed points within the ellipse. For an ellipse with the major axis along the x-axis, the coordinates of the foci are $(\pm ae, 0)$.

4. Focal Property of an Ellipse: For any point $P$ on the ellipse, the sum of its distances from the two foci is constant and equal to $2a$. That is, $SP + S^{\prime}P = 2a$. This is the defining characteristic of an ellipse.

5. Focal Distances ($SP$ and $S^{\prime}P$): The distances from a point $P(x,y)$ on the ellipse to the foci $S(ae,0)$ and $S'(-ae,0)$ can be directly expressed in terms of $a$, $e$, and the x-coordinate of $P$:

   • $SP = a - ex$
   • $S^{\prime}P = a + ex$ These formulas are derived from the definition of an ellipse as the locus of points whose ratio of distance from a focus to distance from a directrix is $e$. They are incredibly useful for problems involving focal distances.

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Step-by-Step Solution

Step 1: Identify the parameters of the given ellipse.

The given equation of the ellipse is $\frac{x^2}{18} + \frac{y^2}{9} = 1$. We compare this with the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

• From the equation, we have $a^2 = 18$ and $b^2 = 9$.
• Since $18 > 9$, we have $a^2 > b^2$, which confirms that the major axis lies along the x-axis, and our chosen standard formulas are appropriate.
• Taking the square root, we get $a = \sqrt{18} = 3\sqrt{2}$ and $b = \sqrt{9} = 3$.

Step 2: Calculate the eccentricity ($e$) and the product $ae$.

To find the eccentricity, we use the relation $b^2 = a^2(1-e^2)$. Substituting the values of $a^2$ and $b^2$: $9 = 18(1-e^2)$ $1 - e^2 = \frac{9}{18} = \frac{1}{2}$ $e^2 = 1 - \frac{1}{2} = \frac{1}{2}$ $e = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$ Now, we calculate $ae$, which is the distance from the center to each focus: $ae = (3\sqrt{2}) \cdot \left(\frac{1}{\sqrt{2}}\right) = 3$ The coordinates of the foci are $S(ae, 0) = (3,0)$ and $S'(-ae, 0) = (-3,0)$. (While not strictly needed for this problem, it's good practice to identify them).

Step 3: Express the product $SP \cdot S'P$ in terms of $a$, $e$, and $x$.

Let $P(x,y)$ be any point on the ellipse. We use the direct formulas for focal distances:

• $SP = a - ex$
• $S^{\prime}P = a + ex$ Now, we find their product: $SP \cdot S^{\prime}P = (a - ex)(a + ex)$ This is a difference of squares: $SP \cdot S^{\prime}P = a^2 - (ex)^2$ $SP \cdot S^{\prime}P = a^2 - e^2x^2$ This expression simplifies the problem significantly, as we now only need to find the range of $x$ for a point on the ellipse.

Step 4: Determine the range of $x$ for a point $P(x,y)$ on the ellipse.

For any point $P(x,y)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the x-coordinate must satisfy $-a \le x \le a$. From Step 1, we found $a = 3\sqrt{2}$. Therefore, for any point $P$ on the ellipse, its x-coordinate $x$ lies in the interval: $-3\sqrt{2} \le x \le 3\sqrt{2}$ This means $0 \le x^2 \le (3\sqrt{2})^2 = 18$.

Step 5: Find the minimum and maximum values of $SP \cdot S'P$.

We have the expression $SP \cdot S^{\prime}P = a^2 - e^2x^2$. Substitute the values of $a^2$ and $e^2$: $SP \cdot S^{\prime}P = 18 - \left(\frac{1}{2}\right)x^2$ Let $K = SP \cdot S^{\prime}P$. We want to find $\min(K)$ and $\max(K)$.

• To maximize $K$: We need to minimize the term $e^2x^2$ (or $\frac{1}{2}x^2$). This occurs when $x^2$ is at its minimum value, which is $x^2 = 0$. This happens when $x=0$, meaning the point $P$ is at $(0, \pm b)$, which are the endpoints of the minor axis. $\max(K) = 18 - \frac{1}{2}(0) = 18$ So, $\max(SP \cdot S^{\prime}P) = 18$.

• To minimize $K$: We need to maximize the term $e^2x^2$ (or $\frac{1}{2}x^2$). This occurs when $x^2$ is at its maximum value, which is $x^2 = a^2 = 18$. This happens when $x = \pm a = \pm 3\sqrt{2}$, meaning the point $P$ is at $(\pm a, 0)$, which are the endpoints of the major axis. $\min(K) = 18 - \frac{1}{2}(18) = 18 - 9 = 9$ So, $\min(SP \cdot S^{\prime}P) = 9$.

Step 6: Calculate the sum $\min \left(S P \cdot S^{\prime} P\right)+\max \left(S P \cdot S^{\prime} P\right)$.

Finally, we sum the minimum and maximum values we found: $\min \left(S P \cdot S^{\prime} P\right)+\max \left(S P \cdot S^{\prime} P\right) = 9 + 18 = 27$

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Tips and Common Mistakes to Avoid:

1. Correctly identify $a^2$ and $b^2$: Always remember that $a^2$ is the larger denominator and $b^2$ is the smaller denominator for an ellipse, irrespective of whether it's under $x^2$ or $y^2$. This determines whether the major axis is horizontal or vertical. In this problem, $a^2=18$ and $b^2=9$.
2. Memorize or derive focal distance formulas: The formulas $SP = a \pm ex$ are extremely powerful and simplify calculations involving focal distances. Trying to use the distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ directly from $P(x,y)$ to the foci would be much more complicated and error-prone.
3. Understanding the range of $x$: The x-coordinate of any point on the ellipse always lies between $-a$ and $a$. This understanding is key to finding the min/max values of expressions dependent on $x$.
4. Minimizing/Maximizing $a^2 - e^2x^2$: To maximize an expression of the form $C - D \cdot X$, you need to minimize $X$. To minimize it, you need to maximize $X$. This is a common optimization technique.

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Summary and Key Takeaway

This problem elegantly combines several key properties of an ellipse. The most crucial steps were:

1. Correctly identifying the semi-major axis $a$ and semi-minor axis $b$ from the ellipse equation.
2. Calculating the eccentricity $e$.
3. Using the simplified focal distance formulas $SP = a - ex$ and $S^{\prime}P = a + ex$ to express the product $SP \cdot S^{\prime}P$ as $a^2 - e^2x^