Step-by-step Solution:

To find the correct value of \( a \), we proceed as follows: Step 1: Find the slope of the tangent to \( ay + x^2 = 7 \) at \( (1,1) \) Differentiate both sides with respect to \( x \): \[ a \frac{dy}{dx} + 2x = 0 \] Substituting \( x = 1 \): \[ a \frac{dy}{dx} + 2(1) = 0 \] \[ a \frac{dy}{dx} = -2 \] \[ \frac{dy}{dx} = -\frac{2}{a} \] Thus, the slope of the tangent at \( (1,1) \) is: \[ m_1 = -\frac{2}{a} \] --- Step 2: Find the slope of the tangent to \( x^3 = y \) at \( (1,1) \) Differentiate both sides with respect to \( x \): \[ 3x^2 = \frac{dy}{dx} \] Substituting \( x = 1 \): \[ 3(1)^2 = \frac{dy}{dx} \] \[ \frac{dy}{dx} = 3 \] Thus, the slope of the tangent at \( (1,1) \) is: \[ m_2 = 3 \] Step 3: Use the orthogonality condition For the curves to cut orthogonally: \[ m_1 \times m_2 = -1 \] \[ \left(-\frac{2}{a}\right) \times 3 = -1 \] \[ -\frac{6}{a} = -1 \] \[ a = 6 \] Final Answer: \[ \boxed{6} \]