Let P Q be a chord of the parabola y²=12 x and the midpoint of P Q be at (4,1). Then, which of the following point lies on the line passing through the points P and Q ?

Solution

Maths sir · 2 steps

Subtract the two parabola equations for P and Q, and the midpoint hands you the slope of the chord.

1. Slope from midpoint. Let P(x₁, y₁) and Q(x₂, y₂) lie on y² = 12x. Subtracting gives y₁² - y₂² = 12(x₁ - x₂), so the slope is (y₁ - y₂)/(x₁ - x₂) = 12/(y₁ + y₂). The midpoint is (4, 1), so y₁ + y₂ = 2 and the slope is 6.

   y - 1 = 6(x - 4) ⇒ y = 6x - 23

   Point	6x - 23	Given y	On the line?
   (a) (3, -3)	-5	-3	No
   (b) (1/2, -20)	-20	-20	Yes
   (c) (2, -9)	-11	-9	No
   (d) (3/2, -16)	-14	-16	No

2. So. Only (1/2, -20) satisfies y = 6x - 23, so the answer is (b).

The idea to keep. For a chord of y² = 4ax with midpoint (h, k), the slope is 2a/k. You can also write the chord directly using T = S₁.

Answer from the NTA final answer key. Solution by PrepWiser.