Question

If P is a point on the parabola y = x² + 4 which is closest to the straight line y = 4x - 1, then the co-ordinates of P are :

• A. (3, 13)
• B. (2, 8)
• C. (-2, 8)
• D. (1, 5)

Hint:

Shortest distance between a non-intersecting line and a curve always lies along the common normal.

Answer 1

The Correct Option is B

Step 1: The closest point on a curve to a line is where the tangent to the curve is parallel to the line.
Step 2: Slope of the line $y = 4x - 1$ is $m = 4$.
Step 3: Differentiate parabola: $\frac{dy}{dx} = 2x$. Set $\frac{dy}{dx} = 4$.
Step 4: $2x = 4 \Rightarrow x = 2$.
Step 5: Substitute $x = 2$ into parabola equation: $y = (2)^2 + 4 = 8$. Point $P$ is $(2, 8)$.