Question

Let the point $ P $ of the focal chord $ PQ $ of the parabola $ y^2 = 16x $ be $ (1, -4) $. If the focus of the parabola divides the chord $ PQ $ in the ratio $ m : n $, gcd($m, n$) = 1, then $ m^2 + n^2 $ is equal to:

• A. 17
• B. 10
• C. 37
• D. 26

Hint:

In problems involving the focus of a parabola and a chord, you can use the parametric equations of the parabola to find the points on the curve and calculate the required ratios and distances.

Answer 1

The Correct Option is A

The equation of the parabola is: \[ y^2 = 16x; \, a = 4 \] The focus \( S \) is \( (4, 0) \), and the point \( P \) is \( (1, -4) \). 
From the equation of the parabola, we know the parametric equations for the points on the parabola: \[ t_1 = -4, \, 2a t_1 = -4 \implies t_1 = \frac{-1}{2} \] \[ t_2 = 2 \implies Q(at_2^2, 2at_2) = (16, 16) \] Let \( S \) divides \( PQ \) internally in the ratio \( \lambda : 1 \): \[ 16\lambda - 4 = 0 \quad \implies \lambda = \frac{1}{4} \] Thus, the ratio \( \frac{m}{n} = \frac{1}{4} \), and: \[ m^2 + n^2 = 1 + 16 = 17 \]