Question

In the parabola $y^2 = 4ax$ the length of the latus rectum is 6 units and there is a chord passing through its vertex and the negative end of the latus rectum. Then the equation of the chord is

• A. \( x + 2y = 0 \)
• B. \( 2x + y = 0 \)
• C. \( x - 2y = 0 \)
• D. \( 2x - y = 0 \)

Hint:

In parabolas of the form \( y^2 = 4ax \), the equation of a chord passing through the vertex and the end of the latus rectum can be derived by applying the general property of the parabola.

Answer 1

The Correct Option is B

We are given the equation of the parabola: \[ y^2 = 4ax. \] The length of the latus rectum for this parabola is given by \( 4a \), and it is given that the length of the latus rectum is 6 units, so: \[ 4a = 6 \quad \Rightarrow \quad a = \frac{3}{2}. \] The equation of the chord passing through the vertex and the negative end of the latus rectum can be determined using the general property of the parabola. The negative end of the latus rectum is located at the point \( \left( -a, 0 \right) \), which for this parabola is \( \left( -\frac{3}{2}, 0 \right) \). For a parabola of the form \( y^2 = 4ax \), the equation of the chord through the vertex and the end of the latus rectum can be written as: \[ 2x + y = 0. \] Thus, the equation of the chord is \( 2x + y = 0 \). Hence, the correct answer is (B).