Question

Point A and B of a triangle ABC are at (-1,1) and (3,9), respectively; while point C is on the parabola y=x2. Find coordinate of C so that the area of ABC is minimum, and calculate the largest are of ABC.

Answer 1

Let AB be the base, which will be a fixed length
AB = √(4^2 + 8^2) = √80

The height from C to line AB must be as long as possible, this is achieved when the slope of the tangent at C is the same as the slope of AB,

dy/dx = 2x
slope of AB = 8/4 = 2

then 2x = 2
x = 1

when x = 1, y = 1^2 = 1

C must be (1,1)