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 parabola's equation is \( y^2 = 16x \), with \( a = 4 \). The focus \( S \) is at \( (4, 0) \), and point \( P \) is at \( (1, -4) \).
From the parabola's equation, its parametric form is derived. For point \( P \), \( t_1 \) is calculated as \( -4 \), and \( 2a t_1 = -4 \), which yields \( t_1 = \frac{-1}{2} \). For point \( Q \), \( t_2 = 2 \), and \( Q(at_2^2, 2at_2) = (16, 16) \). Assuming \( S \) divides \( PQ \) internally in the ratio \( \lambda : 1 \), we have \( 16\lambda - 4 = 0 \), leading to \( \lambda = \frac{1}{4} \). Therefore, the ratio \( \frac{m}{n} = \frac{1}{4} \), and \( m^2 + n^2 = 1 + 16 = 17 \).