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 \( 4\pi \).

Answer 2

Given:
- Point \( A = (4, 4) \)
- Point \( Q \) lies on the circle:
\[ x^2 + y^2 = 9 \] - Point \( P \) divides segment \( QA \) internally in ratio 1:2.

Step 1: Let \( Q = (x, y) \) on the circle \( x^2 + y^2 = 9 \).
Step 2: Coordinates of \( P \), dividing \( Q A \) in ratio 1:2 internally:
\[ P = \left( \frac{1 \cdot 4 + 2 \cdot x}{1 + 2}, \frac{1 \cdot 4 + 2 \cdot y}{1 + 2} \right) = \left( \frac{4 + 2x}{3}, \frac{4 + 2y}{3} \right) \]

Step 3: Find the locus of \( P \) by eliminating \( x, y \).
Express \( x, y \) in terms of \( P = (X, Y) \):
\[ X = \frac{4 + 2x}{3} \implies 3X = 4 + 2x \implies 2x = 3X - 4 \implies x = \frac{3X - 4}{2} \] \[ Y = \frac{4 + 2y}{3} \implies y = \frac{3Y - 4}{2} \]

Step 4: Since \( Q = (x, y) \) lies on the circle:
\[ x^2 + y^2 = 9 \] Substitute \( x, y \):
\[ \left( \frac{3X - 4}{2} \right)^2 + \left( \frac{3Y - 4}{2} \right)^2 = 9 \] Multiply both sides by 4:
\[ (3X - 4)^2 + (3Y - 4)^2 = 36 \] Expand:
\[ 9 X^2 - 24 X + 16 + 9 Y^2 - 24 Y + 16 = 36 \] \[ 9 X^2 + 9 Y^2 - 24 X - 24 Y + 32 = 36 \] \[ 9 X^2 + 9 Y^2 - 24 X - 24 Y = 4 \]

Step 5: Divide entire equation by 9:
\[ X^2 + Y^2 - \frac{24}{9} X - \frac{24}{9} Y = \frac{4}{9} \] \[ X^2 + Y^2 - \frac{8}{3} X - \frac{8}{3} Y = \frac{4}{9} \]

Step 6: Complete the squares:
\[ X^2 - \frac{8}{3} X + Y^2 - \frac{8}{3} Y = \frac{4}{9} \] Add and subtract \( \left( \frac{4}{3} \right)^2 = \frac{16}{9} \) for both \( X \) and \( Y \):
\[ \left( X^2 - \frac{8}{3} X + \frac{16}{9} \right) + \left( Y^2 - \frac{8}{3} Y + \frac{16}{9} \right) = \frac{4}{9} + \frac{16}{9} + \frac{16}{9} \] \[ \left( X - \frac{4}{3} \right)^2 + \left( Y - \frac{4}{3} \right)^2 = \frac{36}{9} = 4 \]

Step 7: The locus of \( P \) is a circle with center \( \left( \frac{4}{3}, \frac{4}{3} \right) \) and radius \( 2 \).

Step 8: The perimeter of the locus (circumference of the circle) is:
\[ 2 \pi \times 2 = 4 \pi \]

Therefore, the perimeter is:
\[ \boxed{4 \pi} \]