Question

If the locus of a point that divides a chord of slope 2 of the parabola \( y^2 = 4x \) internally in the ratio 1 : 2 is a parabola, then its vertex is

• A. \( \left( \frac{2}{9}, \frac{8}{9} \right) \)
• B. \( \left( \frac{1}{3}, \frac{9}{9} \right) \)
• C. \( \left( \frac{4}{9}, \frac{8}{9} \right) \)
• D. \( \left( \frac{2}{9}, \frac{4}{9} \right) \)

Hint:

To find the locus of a point dividing a chord of a conic, parametrize the endpoints, apply the section formula, and eliminate the parameter.

Answer 1

The Correct Option is A

Step 1: Understand the geometry. We are given a parabola \( y^2 = 4x \) and a chord with slope 2. A point divides this chord in the ratio \( 1:2 \) internally. The locus of such a point is another parabola. Step 2: Equation of chord with given slope. Let the endpoints of the chord be \( A(x_1, y_1) \) and \( B(x_2, y_2) \) such that both lie on the parabola. The point dividing \( AB \) in \( 1:2 \) internally is: \[ \left( \frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3} \right) \] Step 3: Use geometry and symmetry of parabola. Using the method of parametric coordinates \( (at^2, 2at) \) for \( y^2 = 4ax \) where \( a = 1 \), we take two points on the parabola and form the chord with slope 2. After algebraic substitution and simplification, we find the locus of the dividing point, and from that the vertex of the resulting locus parabola is: \[ \left( \frac{2}{9}, \frac{8}{9} \right) \]