Question

The perimeter of the locus of the point \( P \) which divides the line segment \( QA \) internally in the ratio 1:2, where \( A = (4, 4) \) and \( Q \) lies on the circle \( x^2 + y^2 = 9 \), is:

• A. \( 8\pi \)
• B. \( 4\pi \)
• C. \( \pi \)
• D. \( 9\pi \)

Hint:

When dividing a line segment in a given ratio, use the section formula to find the coordinates of the dividing point and analyze the resulting locus.

Answer 1

The Correct Option is B

We are given that \( Q \) lies on the circle \( x^2 + y^2 = 9 \), and the point \( P \) divides the line segment \( QA \) in the ratio 1:2.
First, we determine the parametric equations for the coordinates of \( P \). 
The coordinates of \( P \) that divides \( QA \) in the ratio \( 1:2 \) are given by the section formula: \[ P = \left( \frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3} \right), \] where \( A = (4, 4) \) and \( Q = (x_1, y_1) \) lies on the circle \( x^2 + y^2 = 9 \). So, \( P \) will trace a curve as \( Q \) moves along the circle. Since \( P \) divides \( QA \) in the ratio 1:2, the locus of \( P \) will be a circle with radius \( \frac{2}{3} \) of the radius of the original circle. 
The radius of the circle traced by \( P \) is \( \frac{2}{3} \times 3 = 2 \). 
Thus, the perimeter (circumference) of the locus of \( P \) is: \[ \text{Perimeter} = 2\pi \times 2 = 4\pi. \] Thus, the correct answer is \( 4\pi \).