Mathematics

Problem

Consider two curves

y^2 = 4a(x - \lambda)

and

x^2 = 4a(y - \lambda)

, where

a > 0

and

\lambda

is a parameter. Show that a. there is a single positive value of

\lambda

for which the two curves have exactly one point of intersection in the 1st quadrant find it. b. there are infinitely many negative values of

\lambda

for which the two curves have exactly one point of intersection in the 1st quadrant. c. if

\lambda = -a

, then find the area of the bounded by the two curves and the axes in the 1st quadrant.