Question

r(x)=(x-a)^2+(x+a)^2

Answer 1

r(x) = (x + a)² + (x - a)²
= x² + a² + 2ax + x² + a² - 2ax
= 2(x² + a²)

this is the equation of parabola .
in which

focus : ( 0, 2a² + 1/8)
vertex : ( 0, 2a²)
semi - axis length : 1/8
focal parameter : 1/4
eccentricity : 1
directrix : y = 2a² - 1/8

see the figure , I gave 3D graph of given polynomial .