Key Concepts and Formulas

• Power of a Point Theorem: If a line through a point P intersects a circle at points A and B, then PA ⋅ PB is constant for any such line through P.
• Equation of a Circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. The power of a point $(x_1, y_1)$ with respect to this circle is $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$.
• Circle Equation (Simplified): For a circle $x^2 + y^2 = r^2$, the power of a point $(x_1, y_1)$ is $x_1^2 + y_1^2 - r^2$.

Step-by-Step Solution

Step 1: Identify the given information We are given the point P(4, 7) and the equation of the circle $x^2 + y^2 = 9$. We want to find the value of $PA \cdot PB$.

Step 2: Rewrite the circle equation in the standard form The equation of the circle is already in a simplified standard form, $x^2 + y^2 = r^2$, where $r^2 = 9$. We can rewrite it as $x^2 + y^2 - 9 = 0$.

Step 3: Apply the Power of a Point Theorem According to the Power of a Point Theorem, $PA \cdot PB$ is equal to the power of the point P with respect to the circle. Let $S(x, y) = x^2 + y^2 - 9$. The power of the point P(4, 7) is $S(4, 7)$.

Step 4: Calculate the power of the point Substitute the coordinates of point P into the equation $S(x, y) = x^2 + y^2 - 9$: $S(4, 7) = (4)^2 + (7)^2 - 9$ $S(4, 7) = 16 + 49 - 9$ $S(4, 7) = 65 - 9$ $S(4, 7) = 56$

Step 5: Relate the result to the problem The power of the point P with respect to the circle is 56. Therefore, $PA \cdot PB = 56$.

Common Mistakes & Tips

• Ensure the circle equation is in the form $x^2 + y^2 + 2gx + 2fy + c = 0$ before substituting the point's coordinates. This often means moving the constant term to the left side of the equation.
• Double-check your arithmetic to avoid simple calculation errors.

Summary

The problem asks for $PA \cdot PB$, which is the power of the point P(4, 7) with respect to the circle $x^2 + y^2 = 9$. By substituting the coordinates of P into the equation $x^2 + y^2 - 9 = 0$, we find that $PA \cdot PB = 56$.

The final answer is $\boxed{56}$, which corresponds to option (B).