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: The equations of the lines form a triangle, and we need to find the coordinates of the centroid of the triangle. To do this, solve the system of linear equations to find the vertices \( A \), \( B \), and \( C \) of the triangle. 
Step 2: The centroid of a triangle is the average of the coordinates of its vertices. After determining the coordinates of the vertices, calculate the centroid using the formula: \[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] 
Step 3: Next, find the image of the centroid under the given line transformation. The image of the centroid \( G \) is the point \( (h, k) \). 
Step 4: Finally, calculate \( h^2 + k^2 + hk \) using the obtained values of \( h \) and \( k \). Thus, the correct answer is (4).