Let the tangent to the curve x²+2 x-4 y+9=0 at the point P(1,3) on it meet the y-axis at A. Let the line passing through P and parallel to the line x-3 y=6 meet the parabola y²=4 x at B. If B lies on the line 2 x-3 y=8, then (AB)² is equal to ___________.

Solution

Maths sir · 4 steps

Find A from the tangent, then find B from two meeting points, and let the extra line choose between them.

1. Tangent at P. Differentiate x² + 2x - 4y + 9 = 0 to get 2x + 2 - 4dy/dx = 0, so dy/dx = (x+1)/2 = 1 at x = 1. The tangent is y - 3 = x - 1, that is y = x + 2. Putting x = 0 gives A(0, 2).
2. Line through P. The line x - 3y = 6 has slope 1/3. The parallel line through P(1, 3) is y - 3 = 1/3(x - 1), that is x = 3y - 8. Substitute this into y² = 4x.

   y² = 4(3y - 8) ⇒ y² - 12y + 32 = 0 ⇒ y = 4 or y = 8

3. Pick B. The two meeting points are (4, 4) and (16, 8). Test them in 2x - 3y = 8. For (4, 4) you get 8 - 12 = -4, so it fails. For (16, 8) you get 32 - 24 = 8, so it works. So B = (16, 8).
4. So. (AB)² = (16 - 0)² + (8 - 2)² = 256 + 36 = 292.

The common slip: 20. Taking the other meeting point (4, 4) as B gives (AB)² = 16 + 4 = 20. That point does not lie on 2x - 3y = 8.

The idea to keep. A line meets a parabola in two points, so the extra condition in the question is there to pick one of them. Always test both roots before you calculate any distance.

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