let C1 : y = x - 1/2 x2 and C2 : x = y - 1/2 y2 be curves on the xy plane.
1. find the equation of the tangent to the curve C1 at x = k
2. suppose the line obtained in 1) is also tangent to the curve C2. find all values of k and the equations of the tangents.
3. evaluate the area of the figure enclosed by all tangents obtained in 2) and the curve C2
derivatives, equation of tangent
The Attempt at a Solution
1. I've done it. I got : y = (1-k) x + 1/2 k2
differentiate C2 with respect to x :
1 = dy/dx - y dy/dx
dy/dx = 1/(1-y) = m
so, 1/(1-y) = 1-k