Question

If \( \left(\frac{2}{3},0\right) \) is the centroid of the triangle formed by the lines \( 4x^2 - y^2 = 0 \) and \( lx + my + n = 0 \), then \( l+m+n= \):

• A. \( 1 \)
• B. \( -1 \)
• C. \( 2 \)
• D. \( 2 \)

Hint:

When dealing with centroid calculations for triangles formed by intersecting lines, always use: \[ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] and solve accordingly.

Answer 1

The Correct Option is A

The given equation \( 4x^2 - y^2 = 0 \) represents a pair of straight lines passing through the origin. The centroid of a triangle formed by three intersecting lines is given by: \[ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] Solving for the centroid condition: \[ \frac{x_1 + x_2 + x_3}{3} = \frac{2}{3}, \frac{y_1 + y_2 + y_3}{3} = 0 \] Solving for \( l, m, n \), we get: \[ l + m + n = 1 \]