Question

Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0 \), \( x + 2y - 31 = 0 \), and \( 9x - 2y - 19 = 0 \). 
Let the point \( (h, k) \) be the image of the centroid of \( \triangle ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to:

• A. 47
• B. 36
• C. 40
• D. 37

Hint:

When solving for the centroid of a triangle, use the formula for the centroid and then apply any given transformations to find the image of the point.

Answer 1

The Correct Option is D

Step 1: Determine the vertices \( A \), \( B \), and \( C \) of the triangle formed by the given lines by solving the system of linear equations.

Step 2: Calculate the centroid \( G \) of the triangle using the average of the vertex coordinates: \[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Step 3: Find the image of the centroid \( G \) under the specified line transformation, denoted as \( (h, k) \).

Step 4: Compute the value of \( h^2 + k^2 + hk \) using the determined coordinates \( h \) and \( k \). The correct answer is (4).