If the tangent to the curve, y = eˣ at a point (c, eᶜ) and the normal to the parabola, y² = 4x at the point (1, 2) intersect at the same point on the x-axis, then the value of c is ________ .

Solution

Maths sir · 4 steps

Find where the normal hits the x-axis first, then make the tangent to eˣ pass through that point.

1. Normal to parabola. From y² = 4x, 2y dy/dx = 4, so dy/dx = 2/y = 1 at (1, 2). The normal has slope -1, so its equation is y - 2 = -(x - 1), that is, x + y = 3.
2. Meeting the x-axis. Put y = 0 in x + y = 3. The normal cuts the x-axis at (3, 0).
3. Tangent to exponential. For y = eˣ the slope at x = c is eᶜ, so the tangent is y - eᶜ = eᶜ(x - c). Putting y = 0 gives -eᶜ = eᶜ(x - c).

   x - c = -1 ⇒ x = c - 1

4. So. Both lines meet the x-axis at the same point, so c - 1 = 3, which gives c = 4.

The common slip: 2. Dividing -eᶜ = eᶜ(x - c) with the sign lost gives x = c + 1, so c + 1 = 3 and c = 2. Keep the minus sign: x - c = -1.

The idea to keep. The tangent to y = eˣ at x = c always cuts the x-axis at x = c - 1. Remember this result; it saves a full step whenever eˣ tangents show up.

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