Question

Eliminate a and B from the equation Z=(x^2+a)(y^2+b)

Answer 1

$$z=(x^2+a)(y^2+b)=x^2y^2+ay^2+bx^2+ab$$

Differentiate $z$ with respect to $x$

$$p=\dfrac{\partial z}{\partial x}=2xy^2+2bx$$

Differentiate $z$ with respect to $y$

$$q=\dfrac{\partial z}{\partial y}=2x^2y+2ay$$

Then

$$pq=(2xy^2+2bx)(2x^2y+2ay)$$

$$=4xy(y+b)(x+a)=4xyz$$

Therefore

$$pq=4xyz$$