We have received two good solutions to this problem.

Biren Patel (aged 14) from The Heathland School approached it like this:

The question asks you to prove that a+b=c.

You already know that the two lines are parallel.This tells us that they have the same gradient.

To work out the gradients we should try to imagine we draw a right angled triangle under both lines.

We then divide the change in the y-axis by the change in the x-axis.

The change in the x-axis of the line AB is equal to b-a (small letters refer to the co-ordinates). The change in the y-axis is equal to

b2 − a2

(these letters also refer to the co-ordinates).

Thus the gradient of line AB is equal to:

b2 − a2 b − a =  (b−a)(b+a) b−a = b+a

The change in the y-axis of the line OC is  c². The change in the x-axis is equal to c. Thus the gradient of the line OC is

c2 c = c

We know that both lines, AB and OC are parallel, and so they must have the same gradient.

So b+a = c

Andrei Lazanu (aged 12) from School No. 205 in Bucharest (Romania), approached it in a slightly different way:

Let y = mx + n be the equation of the line going through the points A(a, a ²) and B(b, b ²).

At A,

a2 = ma + n

At B,

b2 = mb + n

From this:

a2 − b2 = ma − mb = m (a − b)

Dividing by (a-b):

m =  a2 − b2 a − b =  (a + b)(a − b) a + b = a − b

The line OC passes through the origin and the point C(c, c ²).

Because it passes through the origin, its equation is of the form:

y = ux

Because it is parallel to the line passing through A and B, u = m, so its equation is:

y = (a + b)x

Since it passes through C:

c2 = (a + b)c

or:

c = a + b

Well done to you both.