Question

Let \( C \) be the centre and \( A \) be one end of a diameter of the circle \( x^2 + y^2 - 2x - 4y - 20 = 0 \). If \( P \) is a point on \( AC \) such that \( CP : PA = 2 : 3 \), then the locus of \( P \) is:

• A. \( x^2 + y^2 - 2x - 4y - 205 = 0 \)
• B. \( 2x^2 + 2y^2 - 4x - 8y - 405 = 0 \)
• C. \( x^2 + y^2 - 2x - 4y - 450 = 0 \)
• D. \( 4x^2 + 4y^2 - 8x - 16y - 605 = 0 \)

Hint:

When a point divides a line segment in a given ratio, use the section formula to find its coordinates.

Answer 1

The Correct Option is D

Step 1: Rewrite the equation of the circle.
The equation of the circle is \( x^2 + y^2 - 2x - 4y - 20 = 0 \), which becomes \( (x - 1)^2 + (y - 2)^2 = 25 \).
Step 2: Find the coordinates of \( A \).
The coordinates of \( A \) are \( (-4, 2) \).
Step 3: Use the section formula.
Substituting \( m = 2 \) and \( n = 3 \), the coordinates of \( P \) are \( (-2, 2) \). Final Answer: \[ \boxed{4x^2 + 4y^2 - 8x - 16y - 605 = 0} \]