MathDB

Consider the parabolas C_a: y \equal{} \frac {1}{4}x^2 \minus{} ax \plus{} a^2 \plus{} a \minus{} 2 and C: y \equal{} \minus{} \frac {1}{4}x^2 \plus{} 2 for real number

a

in the x \minus{} y plane. (1) Find the equation of the locus of the vertex for

C_a.

(2) For a \equal{} 3, find the slope of the two common tangent lines of

C

and

C_a,

then the intersection points of the lines. (3) Suppose that

C_a

intersects with

C

at two distinct points. Find the maximum area of the figure bounded by

C

and

C_a.

Problem Statement