A tangent is drawn to the parabola y² = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?

Solution

Maths sir · 3 steps

Get the tangent from y = mx + a/m, then put each point into it and see which one fails.

1. Find the slope. The line 2x + y = 1 has slope -2. A line perpendicular to it has slope m = 1/2, since (-2) × 1/2 = -1.
2. Write the tangent. Compare y² = 6x with y² = 4ax: 4a = 6, so a = 3/2. The tangent with slope m is y = mx + a/m, and here a/m = (3/2)/(1/2) = 3.

   y = x/2 + 3 x - 2y + 6 = 0

   Point	x - 2y + 6	On the tangent?
   (a) (0, 3)	0 - 6 + 6 = 0	yes
   (b) (-6, 0)	-6 - 0 + 6 = 0	yes
   (c) (4, 5)	4 - 10 + 6 = 0	yes
   (d) (5, 4)	5 - 8 + 6 = 3	no

3. So. Only (5, 4) does not satisfy x - 2y + 6 = 0, so the answer is (d).

The idea to keep. For y² = 4ax, the tangent with slope m is y = mx + a/m. Find m from the given condition first, and remember a is a quarter of the coefficient of x.

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