Key Concepts and Formulas

• Equation of an Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively.
• Parametric Coordinates of an Ellipse: A point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ can be represented as $(a \cos \theta, b \sin \theta)$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step-by-Step Solution

Step 1: Convert the Ellipse Equation to Standard Form and Parametrize Point P

We are given the equation of the ellipse as $4x^2 + 5y^2 = 20$. To apply parametric coordinates, we need the standard form of the equation.

Divide the equation by 20: $\frac{4x^2}{20} + \frac{5y^2}{20} = 1$ $\frac{x^2}{5} + \frac{y^2}{4} = 1$

This is now in the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a^2 = 5$ and $b^2 = 4$. Therefore, $a = \sqrt{5}$ and $b = 2$.

Now, we represent a general point P on the ellipse using parametric coordinates $(a \cos \theta, b \sin \theta)$. Substituting the values of $a$ and $b$, the coordinates of point P are: $P(\sqrt{5} \cos \theta, 2 \sin \theta)$ This parametrization simplifies the problem by reducing the number of variables from two ($x, y$) to one ($\theta$).

Step 2: Express the Square of the Distance $PQ^2$

We want to find the distance between point P$(\sqrt{5} \cos \theta, 2 \sin \theta)$ and the fixed point Q$(0, -4)$. We will work with the square of the distance, $PQ^2$, to avoid square roots.

The square of the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $(x_2 - x_1)^2 + (y_2 - y_1)^2$. Therefore, $PQ^2 = (\sqrt{5} \cos \theta - 0)^2 + (2 \sin \theta - (-4))^2$ $PQ^2 = (\sqrt{5} \cos \theta)^2 + (2 \sin \theta + 4)^2$ Working with the squared distance simplifies calculations.

Step 3: Simplify the Expression for $PQ^2$

Expand and simplify the expression for $PQ^2$: $PQ^2 = 5 \cos^2 \theta + (4 \sin^2 \theta + 16 \sin \theta + 16)$ $PQ^2 = 5 \cos^2 \theta + 4 \sin^2 \theta + 16 \sin \theta + 16$

To further simplify, we express everything in terms of $\sin \theta$ using the identity $\cos^2 \theta = 1 - \sin^2 \theta$: $PQ^2 = 5(1 - \sin^2 \theta) + 4 \sin^2 \theta + 16 \sin \theta + 16$ $PQ^2 = 5 - 5 \sin^2 \theta + 4 \sin^2 \theta + 16 \sin \theta + 16$ $PQ^2 = -\sin^2 \theta + 16 \sin \theta + 21$ This simplification makes it easier to find the maximum value.

Step 4: Maximize $PQ^2$ using Completing the Square

Let $f(\theta) = PQ^2 = -\sin^2 \theta + 16 \sin \theta + 21$. To find the maximum value, we complete the square. $PQ^2 = -(\sin^2 \theta - 16 \sin \theta) + 21$ $PQ^2 = -(\sin^2 \theta - 16 \sin \theta + 64 - 64) + 21$ $PQ^2 = -[(\sin \theta - 8)^2 - 64] + 21$ $PQ^2 = -(\sin \theta - 8)^2 + 64 + 21$ $PQ^2 = -(\sin \theta - 8)^2 + 85$

Now, $PQ^2$ is maximized when $(\sin \theta - 8)^2$ is minimized. Since $-1 \le \sin \theta \le 1$, the minimum value of $(\sin \theta - 8)^2$ occurs when $\sin \theta$ is as close to 8 as possible, which is when $\sin \theta = 1$. Therefore, the minimum value of $(\sin \theta - 8)^2$ is $(1 - 8)^2 = (-7)^2 = 49$.

Substituting $\sin \theta = 1$ into the expression for $PQ^2$, we get: $PQ^2 = -(1 - 8)^2 + 85$ $PQ^2 = -49 + 85$ $PQ^2 = 36$ Thus, the maximum value of $PQ^2$ is 36.

Common Mistakes & Tips

• Forgetting the Range of Sine: Remember that $-1 \le \sin \theta \le 1$. Failing to consider this range can lead to incorrect maximization/minimization.
• Working with Distance Instead of Squared Distance: Working with $PQ^2$ simplifies the algebra significantly.
• Correctly Completing the Square: Pay close attention to signs when completing the square.

Summary

We found the maximum value of $PQ^2$ by first expressing the coordinates of point P on the ellipse in parametric form. Then, we wrote $PQ^2$ as a function of $\sin \theta$, simplified the expression, and completed the square to find the maximum value. Considering the range of $\sin \theta$, we found that the maximum value of $PQ^2$ is 36.

Final Answer

The final answer is \boxed{36}, which corresponds to option (A).