Question

Let a focal chord \( 12x + 5y - 27 = 0 \) of the parabola \( y^2 = kx \) intersect the parabola at points \( P \) and \( P' \). If \( S \) is the focus of the parabola, then \( 9(SP + SP') \) = ?

• A. 27
• B. 108
• C. 16 SP.SP\(^1\)
• D. 4 SP.SP\(^1\)

Hint:

In parabolas, focal chords have special properties — the sum of distances to the focus and product of distances are often used in solving problems.

Answer 1

The Correct Option is D

In a parabola, the sum of distances from the focus to the endpoints of a focal chord is constant. Given that the focal chord equation is \( 12x + 5y - 27 = 0 \), we apply the property: \[ SP + SP' = \frac{4a}{\sin \theta} \] Multiplying by 9 and evaluating in terms of the dot product \( SP.SP' \), we obtain: \[ 9(SP + SP') = 4 \cdot SP \cdot SP' \]