Question

Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

• A. \( \frac{343\sqrt{3}}{8} \)
• B. \( \frac{343\sqrt{3}}{8} \)
• C. \( \frac{34\sqrt{3}}{3} \)
• D. \( 17\sqrt{3} \)

Hint:

For problems involving parabolas and focal chords: - Use the standard equation of the parabola and the properties of the focal chord (e.g., the relationship between the point on the parabola and the focus). - Employ geometric properties to find areas, especially when perpendiculars from points on the parabola are involved.

Answer 1

The Correct Option is A

\[ (4, 4\sqrt{3}) \text{ lies on } y^2 = 4ax \] \[ \Rightarrow 48 = 4a \cdot 4 \] \[ 4a = 12 \] \[ \Rightarrow y^2 = 12x \text{ is the equation of the parabola.} \] Now, the parameter for point \( P \) is \( t_1 = \frac{2}{\sqrt{3}} \). Therefore, the parameters for point \( Q \) are \( t_2 = -\frac{\sqrt{3}}{2} \), which gives the coordinates of \( Q \) as \( \left( \frac{9}{4}, -3\sqrt{3} \right) \). \textbf{Area of trapezium PQNM:} \[ \text{Area} = \frac{1}{2} \times MN \times (PM + QN) \] \[ = \frac{1}{2} \times MN \times (PS + QS) \] \[ = \frac{1}{2} \times MN \times PQ \] \[ = \frac{1}{2} \times 7\sqrt{3} \times \frac{49}{4} \] \[ = \frac{343\sqrt{3}}{8}\]