Step 1
Equation of the given curves are
$ ay+x^2=7\: and \: x^3=y$
Consider the equation,
$ ay+x^2=7$
differentiating w.r.t $x$ we get,
$ a.\large\frac{dy}{dx}+2x=0$
$ \Rightarrow \large\frac{dy}{dx}=-\large\frac{2x}{a}$
Let this be $ m_1 \Rightarrow m_1=-\large\frac{2x}{a}$
$ \Rightarrow m_1\: at \: (1,1)=-\large\frac{2}{a}$
Consider the equation,
$y=x^3$
differentiating w.r.t $x$
$ \large\frac{dy}{dx}=3x^2$
Let this be $ m_2 \Rightarrow m_2 \: at \: (1,1)=3$
Step 2
Since the tangents touch orthogonally,
$ m_1 \times m_2 = -1$
$ -\large\frac{2}{a} \times 3 = -1$
$ \Rightarrow a = 6$
The correct option is D