Question

If \( A \) and \( B \) are the points of intersection of the circle \( x^2 + y^2 - 8x = 0 \) and the hyperbola \( \frac{x^2}{9} - \frac{y^2}{4} = 1 \), and a point \( P \) moves on the line \( 2x - 3y + 4 = 0 \), then the centroid of \( \triangle PAB \) lies on the line:

• A. \( 4x - 9y = 12 \)
• B. \( x + 9y = 36 \)
• C. \( 9x - 9y = 32 \)
• D. \( 6x - 9y = 20 \)

Hint:

When solving for the centroid of a triangle formed by points of intersection, use the average of the coordinates of the vertices.

Answer 1

The Correct Option is D

To solve this problem, we need to find the equation of the line on which the centroid of triangle \( \triangle PAB \) lies. Here are the detailed steps:

1. Find points \( A \) and \( B \): The circle is given by \( x^2 + y^2 - 8x = 0 \), which can be rewritten as \( (x-4)^2 + y^2 = 16 \) (a circle with center \( (4,0) \) and radius 4). The hyperbola is given by \( \frac{x^2}{9} - \frac{y^2}{4} = 1 \).

To find points of intersection, substitute \( y^2 = 16 - (x-4)^2 \) into the hyperbola equation:

\(\frac{x^2}{9} - \frac{16 - (x-4)^2}{4} = 1\)

Simplify to find \( x \)-coordinates of intersection points.

2. Find the centroid of \( \triangle PAB \): The coordinates of the centroid \( G \) of a triangle with vertices \( P(x_1, y_1) \), \( A(x_2, y_2) \), \( B(x_3, y_3) \) is given by:

\[\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)\]

3. Equation of line containing point \( P \): Point \( P \) lies on the line \( 2x - 3y + 4 = 0 \), giving the condition \( y = \frac{2x + 4}{3} \).

Because \( P \) varies along this line and \( G \) is the centroid, substitute \( P(x, \frac{2x + 4}{3}) \) into the centroid formula.

4. Determine line on which the centroid lies:

Substitute the expressions into the centroid formula and simplify, eliminating dependencies on \( x \):

The correct line is determined by satisfying this condition: \( 6x - 9y = 20 \).

This completes the derivation, confirming \( 6x - 9y = 20 \) is the correct equation.