Prove 3 distinct points lying on the some curve

1. Sep 22, 2010

zohapmkoftid

1. The problem statement, all variables and given/known data

Given points (p1, q1), (p2, q2), (p3, q3) in the plane with p1, p2, p3 distinct, show that they lie on some curve with equation y = a + bx + cx2.

It should be related to matrix but I have no idea about this question. Could anyone help? Thanks!

2. Relevant equations

3. The attempt at a solution

2. Sep 23, 2010

hunt_mat

One way to examine this question is given these three point you could find out, what a,b, and c is satisfied for this equation. So you know that:
$$\begin{array}{ccc} q_{1} & = & ap_{1}+bp_{1}+cp_{1} \\ q_{2} & = & ap_{2}+bp_{2}+cp_{2} \\ q_{3} & = & ap_{3}+bp_{3}+cp_{3} \end{array}$$
This is a system of linear equations with unknowns a,b and c, the system can be solved via matrix methods.

3. Sep 23, 2010

zohapmkoftid

$$\begin{array}{ccc} q_{1} & = & ap_{1}+bp_{1}+cp_{1}^{2} \\ q_{2} & = & ap_{2}+bp_{2}+cp_{2}^{2} \\ q_{3} & = & ap_{3}+bp_{3}+cp_{3}^{2} \end{array}$$