mathschallenge.net

Problem

The quadratic curve, y = x²bxc, where b,c ^⊕, is drawn and the points of intersection with the axes are labelled, A, B, and C. A circle is drawn through the points A, B, and C, and a fourth point D is created on the y-axis.

Find the co-ordinate of D.

Solution

As the discriminant of the quadratic equation, x²bxc=0, is b²+4c, which is positive, there will be two roots. The product of the roots, α₁α₂=-c, so one root will be negative and one will be positive. Let the three co-ordinates be A (α₁,0), B (α₂,0) and C (0,c). Hence the lengths, AO = -α₁ (as the root will be negative), OB = α₂, and CO = c.

In a circle with two intersecting chords, the product of the segments are equal: AOOB = COOD, therefore, -α₁α₂ = cOD.

As α₁α₂=-c, we get, OD = 1.

That is, the co-ordinate of D is (0,1), which, amazingly, is independent of both b and c.

What if b was allowed to be positive or negative?
What about c?
Investigate the general quadratic curve, y = ax²+bx+c.
Prove the result that in a circle with two intersecting chords, the product of the segments are equal.