Key Concepts and Formulas

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Equation of a Circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $r = \sqrt{g^2 + f^2 - c}$.
• Locus: The set of all points satisfying a given condition.

Step-by-Step Solution

Step 1: Define Points and the Given Condition

Let $P = (x, y)$ be the moving point. Let $A = (5, 0)$ and $B = (-5, 0)$ be the two fixed points. The given condition is that the distance from $P$ to $A$ is three times the distance from $P$ to $B$, i.e., $PA = 3PB$. Squaring both sides to eliminate the square root in the distance formula, we get $PA^2 = 9PB^2$.

Step 2: Apply the Distance Formula

We use the distance formula to express $PA^2$ and $PB^2$ in terms of $x$ and $y$. $PA^2 = (x - 5)^2 + (y - 0)^2 = (x - 5)^2 + y^2$ $PB^2 = (x - (-5))^2 + (y - 0)^2 = (x + 5)^2 + y^2$

Substitute these into the equation $PA^2 = 9PB^2$: $(x - 5)^2 + y^2 = 9[(x + 5)^2 + y^2]$

Step 3: Expand and Simplify the Equation

Expanding the equation, we have: $x^2 - 10x + 25 + y^2 = 9(x^2 + 10x + 25 + y^2)$ $x^2 - 10x + 25 + y^2 = 9x^2 + 90x + 225 + 9y^2$

Rearrange the terms to get: $0 = 8x^2 + 8y^2 + 100x + 200$

Divide by 4 to simplify: $2x^2 + 2y^2 + 25x + 50 = 0$

Divide by 2 to get the standard form: $x^2 + y^2 + \frac{25}{2}x + 25 = 0$

Step 4: Identify the Circle's Parameters

Comparing the equation $x^2 + y^2 + \frac{25}{2}x + 25 = 0$ with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have: $2g = \frac{25}{2}$, so $g = \frac{25}{4}$ $2f = 0$, so $f = 0$ $c = 25$

Step 5: Calculate the Radius

The radius $r$ is given by $r = \sqrt{g^2 + f^2 - c}$. $r^2 = g^2 + f^2 - c = \left(\frac{25}{4}\right)^2 + 0^2 - 25 = \frac{625}{16} - 25 = \frac{625 - 400}{16} = \frac{225}{16}$

Step 6: Calculate 4r^2

We are asked to find the value of $4r^2$. $4r^2 = 4 \times \frac{225}{16} = \frac{225}{4} = 56.25$

Step 7: Round to Nearest Integer

Since the question expects an integer answer, we round $56.25$ to the nearest integer, which is 56. However, since the given correct answer is 3, let's re-evaluate the problem.

The problem states that the locus is a circle of radius r, then find 4r^2. $PA = 3PB$ $PA^2 = 9PB^2$ $(x-5)^2 + y^2 = 9((x+5)^2 + y^2)$ $x^2 - 10x + 25 + y^2 = 9(x^2 + 10x + 25 + y^2)$ $x^2 - 10x + 25 + y^2 = 9x^2 + 90x + 225 + 9y^2$ $8x^2 + 100x + 8y^2 + 200 = 0$ $x^2 + y^2 + \frac{25}{2}x + 25 = 0$ $r^2 = (\frac{25}{4})^2 + 0^2 - 25 = \frac{625}{16} - \frac{400}{16} = \frac{225}{16}$ $r = \frac{15}{4}$ The problem asks for $4r^2$. $4r^2 = 4 * \frac{225}{16} = \frac{225}{4} = 56.25$.

It seems there is an error in the provided correct answer. Let's check if $PA = \frac{1}{3}PB$ instead. $9PA^2 = PB^2$ $9((x-5)^2 + y^2) = (x+5)^2 + y^2$ $9(x^2 - 10x + 25 + y^2) = x^2 + 10x + 25 + y^2$ $9x^2 - 90x + 225 + 9y^2 = x^2 + 10x + 25 + y^2$ $8x^2 - 100x + 8y^2 + 200 = 0$ $x^2 - \frac{25}{2}x + y^2 + 25 = 0$ $r^2 = (\frac{-25}{4})^2 + 0 - 25 = \frac{625}{16} - \frac{400}{16} = \frac{225}{16}$ $4r^2 = \frac{225}{4}$.

If the problem statement was $|PA - PB| = c$, then $c = 2a$, $a = 5$, $c = 10$. If $PA = k PB$, then this is a circle of Apollonius.

If the intended correct answer is 3, then perhaps the question was $4/r^2$. Then $4/r^2 = 4 / (225/16) = 64/225$, which is not 3.

If the intended correct answer is 9, then perhaps the question was $16r^2/25$. $16r^2/25 = 16 * (225/16) / 25 = 9$.

Let's assume the correct answer is 56.25, but the question expects an integer answer, so the intended correct answer is 56.

Based on the given information, the correct answer must be 56.25. Let's assume that $r^2 = \frac{3}{4}$ instead. Then $r^2 = g^2 + f^2 - C = \frac{3}{4}$.

If we are looking for $4r^2 = 3$, then $r^2 = \frac{3}{4}$.

$PA = 3PB$ is the correct equation. The correct value for $4r^2 = \frac{225}{4}$. It is highly likely that the correct answer provided (3) is incorrect.

If $4r^2 = 3$, then $r^2 = \frac{3}{4}$. $\frac{225}{16} = \frac{3}{4}$, which is not possible.

Common Mistakes & Tips

• Be careful with algebraic manipulations, especially when expanding squared terms.
• Remember the correct formula for the radius of a circle in general form: $r = \sqrt{g^2 + f^2 - c}$.
• Double-check your calculations to avoid arithmetic errors.

Summary

The problem asks to find the value of $4r^2$, where $r$ is the radius of the circle that is the locus of a point $P$ such that its distance from $(5, 0)$ is three times its distance from $(-5, 0)$. Following the steps of defining the points, applying the distance formula, simplifying the equation, and calculating the radius, we find that $4r^2 = \frac{225}{4} = 56.25$. Given the problem specifies an integer answer, the correct answer would be 56. However, the problem states that the correct answer is 3, which is wrong.

Final Answer The correct answer is $\boxed{56.25}$. There appears to be an error in the provided correct answer.