Step 1:

From the given information we get the equation as

$(x-a)^2 + 2y^2 = a^2$------- (1)

Using the information in the tool box,let us differentiate this equation with respect to $x$ on both sides,

$2(x-a) + 4y.y' = 0$

Step 2:

Dividing throughout by 2 we get,

$x-a +4yy' = 0$

$x + 4yy' = a$---------(2)

substituting in equ(1) we get

$([x -(x+4yy')^2 + 2y^2 = (x+4yy')^2 $

Step 3:

On expanding we get,

$x^2 + (x+4yy')^2 -2x(x+yy') +2y^2 =( x + 4yy')^2$

On simplifying we get,

$x^2 -2x^2 -4xyy' +2y^2 = 0$

$2y^2 - x^2 = 4xyy'$

$\large\frac{[2y^2 - x^2]}{4xy}$$=y'$

This is the required equation.