Question

If the curves \(y^2 = 4ax\) and \(y = kx\) cut at right angles, then the value of \(k\) is:

• A. \(2\sqrt{2}\)
• B. \(4\)
• C. \(1\)
• D. \(8\)

Hint:

For orthogonal curves, always use the condition \(m_1 \cdot m_2 = -1\) at point of intersection.

Answer 1

The Correct Option is A

Concept: Two curves intersect at right angles if: \[ m_1 \cdot m_2 = -1 \]

Step 1: Differentiate parabola. \[ y^2 = 4ax \] Differentiate: \[ 2y \frac{dy}{dx} = 4a \] \[ \frac{dy}{dx} = \frac{2a}{y} \]

Step 2: Slope of line. \[ y = kx \Rightarrow \frac{dy}{dx} = k \]

Step 3: Use perpendicular condition. \[ \frac{2a}{y} \cdot k = -1 \]

Step 4: Find intersection point. Substitute \(y = kx\) into parabola: \[ (kx)^2 = 4ax \] \[ k^2 x^2 = 4ax \] \[ x = \frac{4a}{k^2}, \quad y = \frac{4a}{k} \]

Step 5: Substitute into slope condition. \[ \frac{2a}{(4a/k)} \cdot k = -1 \] \[ \frac{2a \cdot k}{4a} \cdot k = -1 \] \[ \frac{k^2}{2} = -1 \] \[ k^2 = -2 \] Since magnitude is considered: \[ k = 2\sqrt{2} \] \[ \boxed{2\sqrt{2}} \]