Find the area of triangle in parabola x^2=8yx2=8y?

Let A(x_1, y_1), B(x_2,y_2),C(x_3,y_3)A(x1,y1),B(x2,y2),C(x3,y3) lie on the parabola x^2=8yx2=8y.
Show that the area of DeltaABC is
1/16(x_1-x_2)(x_2-x_3)(x_3-x_1)

1 Answer
Mar 17, 2017

See below.

Explanation:

The area of a triangle given by it's vertices (x_1,y_1),(x_2,y_2),(x_3,y_3) is

A=1/2abs(det((1,1,1),(x_1,x_2,x_3),(y_1,y_2,y_3)))

now substituting y_k=x_k^2/8 we have

A = 1/2det((1,1,1),(x_1,x_2,x_3),(x_1^2/8,x_2^2/8,x_3^2/8))

A = 1/(2cdot 8)det((1,1,1),(x_1,x_2,x_3),(x_1^2,x_2^2,x_3^2))

but

det((1,1,1),(x_1,x_2,x_3),(x_1^2,x_2^2,x_3^2))=(x_1-x_2)(x_2-x_3)(x_3-x_1)

so

A = 1/16 abs((x_1-x_2)(x_2-x_3)(x_3-x_1))