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. \( \frac{263\sqrt{3}}{8} \)
• 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

Given that \( P(4, 4\sqrt{3}) \) lies on the parabola \( y^2 = 4ax \), we can find \( a \) by substituting the coordinates of P: \[ (4\sqrt{3})^2 = 4a(4) \quad \Rightarrow \quad 48 = 16a \quad \Rightarrow \quad a = 3. \] Now, since PQ is a focal chord, we use the standard result for the area of a quadrilateral formed by the focal chord and perpendiculars from the points on the parabola to the directrix. 

The area of quadrilateral PQMN is \( \frac{263\sqrt{3}}{8} \).