Parabella, prove that a +b=c?

Question

take any two points A and B on the parabola y=x^2

Draw the line OC through the origin , parallel to AB, cutting the parabola at C

Let A =(a,a^2), B= (b,b^2), and C=( c,c^2). prove that a+b=c

Answer 1

It's just a case of working out the gradient or slope of each line.

mAB = (b^2 - a^2) / (b-a) = b + a
mOC = c^2 / c = c

Since they are parallel, c = a + b

Answer 2

I assume you mean takting two points that are on the same side of the parabola, otherwise OC only hits the parabola at the origin and C lies ouside of the graph.

Anyway, the slope of line AB is the same as OC, so write the expression for the slopes and set them equal to each other:
(a^2 - b^2)/(a-b) = (c^2 - 0) / (c - 0)
(a+b)(a-b) / (a-b) = c^2 / c
a+b = c

Answer 3

Slope of OC = Slope of AB

Slope of AB = (b^2-a^2)/(b-a)
= (b-a)(b+a)/(b-a)
= (a+b)

A line passes through the origin and (c,c^2) with a slope of (a+b)

y=(a+b)x
y=c^2
x=c
c^2 = (a+b)c
Divide both sides by c
c=a+b
or

a+b=c

Answer 4

The slope of OC must be the same as the slope of AB.
So (b^2 - a^2) / (b - a) = (c^2 - 0) / (c - 0)
<=> [(b - a) (b + a)] / (b - a) = c^2 / c
<=> b + a = c.

Answer 5

Dunno my username is just for the fun.