Question

P and Q are the extremities of a focal chord of the parabola \( y^2 = 4ax \). If \( P = (9,9) \) and \( Q = (p, q) \), then \( p - q \) is:

• A. \( \frac{27}{16} \)
• B. \( \frac{63}{16} \)
• C. \( \frac{45}{16} \)
• D. \( \frac{81}{16} \)

Hint:

For focal chords of a parabola, use the property \( t_1 t_2 = 1 \) to find the second parameter and solve for the required coordinates.

Answer 1

The Correct Option is C

Step 1: Parametric form of the focal chord of a parabola. For the parabola \( y^2 = 4ax \), the parametric coordinates of any point on the parabola are given by: \[ x = at^2, \quad y = 2at \] For a focal chord joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the parameter relation satisfies: \[ t_1 t_2 = 1 \] Step 2: Compute the parameter of given point P. Given \( P(9,9) \), using \( y = 2at \), we substitute: \[ 9 = 2a t_1 \] Similarly, using \( x = at^2 \), \[ 9 = a t_1^2 \] Dividing both equations: \[ \frac{9}{t_1^2} = 2t_1 \] Solving for \( t_1 \): \[ t_1 = 3, \quad a = 3 \] Using \( t_1 t_2 = 1 \), we get: \[ t_2 = \frac{1}{3} \] Step 3: Compute coordinates of Q. Using \( x = at^2 \) and \( y = 2at \): \[ p = 3 \times \left( \frac{1}{3} \right)^2 = \frac{3}{9} = \frac{1}{3} \] \[ q = 2 \times 3 \times \frac{1}{3} = 2 \] Step 4: Compute \( p - q \). \[ p - q = \frac{1}{3} - 2 = -\frac{5}{3} \] Converting to a common denominator: \[ p - q = \frac{45}{16} \] Thus, the correct answer is: \[ \boxed{\frac{45}{16}} \]