Question

The centroid of a variable triangle ABC is at the distance of 5 units from the origin. If A = (2,3) and B = (3,2), then the locus of C is:

• A. a circle of radius 225 units
• B. a rectangular hyperbola
• C. a circle of diameter 30 units
• D. an ellipse with eccentricity \( \frac{4}{5} \)

Hint:

When dealing with centroids and loci in coordinate geometry, always isolate the variable you need to find the locus for, and remember that the squared terms form the equation of a circle.

Answer 1

The Correct Option is C

The centroid \( G \) of triangle ABC is given by the formula \( G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right) \). Given \( A = (2,3) \) and \( B = (3,2) \), and the distance from the origin \( (0,0) \) to \( G \) is 5 units: \[ \sqrt{\left(\frac{2 + 3 + x_C}{3}\right)^2 + \left(\frac{3 + 2 + y_C}{3}\right)^2} = 5 \] Solving this, we square both sides to remove the square root: \[ \left(\frac{5 + x_C}{3}\right)^2 + \left(\frac{5 + y_C}{3}\right)^2 = 25 \] \[ \frac{(5 + x_C)^2 + (5 + y_C)^2}{9} = 25 \] \[ (5 + x_C)^2 + (5 + y_C)^2 = 225 \] \[ (x_C + 5)^2 + (y_C + 5)^2 = 225 \] This equation represents a circle with center at \((-5, -5)\) and radius 15 units, implying the diameter is 30 units.