Equation for C
Construct integrals for area A and B, and then find the equation of curve C.
As point P lies on the curve y=2x2, with x=a, the y co-ordinate of point P is 2a2.
So, let coordinates of point P be (a, 2a2).
To calculate the area A:
It is the area between curve y=2x2 and y=x2, between x=0 and x=a. So
To calculate area B: It is the area between curves C and y=2x2 where y lies between 0 and a2.
It lies in the first quadrant.
Solve y=2x2 for x.
Consider the equation for curve C as x=C(y).
So the area between curves is given by,
Given that areas are equal,
Differentiating with respect to a,
By the first fundamental theorem of calculus and chain rule,
Now, let y=2a2
But, a=y2. So,
Solving the equation for y:
This is the required equation for C