If the line y = 4 + kx, k > 0, is the tangent to the parabola y = x - x² at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

Solution

Maths sir · 4 steps

Put the line into the parabola and make the discriminant zero. That one condition gives both k and P.

1. Tangency condition. Substitute y = 4 + kx into y = x - x² to get x² + (k-1)x + 4 = 0. A tangent meets the parabola at exactly one point, so this quadratic must have equal roots.

   (k-1)² - 16 = 0 ⇒ k - 1 = ± 4 ⇒ k = 5 or k = -3

2. Find P. Since k > 0, take k = 5. The quadratic becomes x² + 4x + 4 = (x+2)² = 0, so x = -2 and y = -2 - 4 = -6. So P = (-2, -6).
3. Find V. Complete the square: y = 1/4 - (x - 1/2)². So the vertex is V = (1/2, 1/4).
4. So. Slope of PV = (1/4 + 6)/(1/2 + 2) = (25/4)/(5/2) = 5/2, which is option (c).

The idea to keep. For a line touching a parabola, substitute and set the discriminant to zero. The repeated root you get is the x-coordinate of the point of contact.

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