Question

The line \( x - 2y - 3 = 0 \) cuts the parabola \( y^2 = 4ax \) at points P and Q. If the focus of this parabola is \( \left(\frac{1}{4}, k\right) \), then PQ is:

• A. \( 16a\sqrt{5} \)
• B. \( 8a\sqrt{5} \)
• C. \( 4a\sqrt{5} \)
• D. \( 2a\sqrt{5} \)

Hint:

For chord length problems in parabolas, use intersection substitution and standard chord length formulas.

Answer 1

The Correct Option is A

Step 1: Finding the intersection points We substitute \( x = \frac{y + 3}{2} \) into the parabola equation \( y^2 = 4ax \): \[ y^2 = 4a \left( \frac{y + 3}{2} \right) \] \[ y^2 - 2ay - 6a = 0 \] Solving this quadratic equation in \( y \) gives the points \( P(y_1) \) and \( Q(y_2) \). Step 2: Finding the distance PQ Using the chord length formula: \[ PQ = \frac{|2a|}{\sqrt{1 + (m^2)}} \] where \( m = 2 \) (slope of line), \[ PQ = \frac{2a \times \sqrt{5}}{1} \] \[ PQ = 16a\sqrt{5} \]