Question

If the line through the point \( P(5,3) \) meets the circle \( x^2 + y^2 - 2x - 4y + \alpha = 0 \) at \( A(4, 2) \) and \( B(x_1, y_1) \), then \( PA \times PB = \):

• A. 6
• B. 12
• C. 9
• D. 8

Hint:

Use the Power of a Point theorem when dealing with intersections of a line through a point with a circle. This theorem provides a quick way to calculate the product of distances from the point to the intersection points.

Answer 1

The Correct Option is D

To solve the problem, we'll use the Power of a Point theorem, which states that for a point \( P \) outside a circle, the product of the lengths of the segments from \( P \) to the points of intersection with the circle is constant. Mathematically, for a point \( P(x_0, y_0) \) and a circle \( x^2 + y^2 + Dx + Ey + F = 0 \), the power of the point is given by: \[ PA \times PB = x_0^2 + y_0^2 + Dx_0 + Ey_0 + F \] Given:
Point \( P(5, 3) \)
Circle equation: \( x^2 + y^2
- 2x
- 4y + \alpha = 0 \)
Point of intersection \( A(4, 2) \)
Step 1: Find the value of \( \alpha \) Since point \( A(4, 2) \) lies on the circle, it satisfies the circle's equation: \[ 4^2 + 2^2
- 2(4)
- 4(2) + \alpha = 0
16 + 4
- 8
- 8 + \alpha = 0
4 + \alpha = 0
\alpha =
-4 \] Step 2: Compute the power of point \( P \) Using the power of a point formula: \[ PA \times PB = 5^2 + 3^2
- 2(5)
- 4(3) + (
-4)
= 25 + 9
- 10
- 12
- 4
= 8 \] Conclusion: The product \( PA \times PB \) is \( 8 \). \boxed{8}