Linear algebra for quadratic curves
Recall that the general equation of a circle of radius r centred at the point (p, q) is (x -p)² + (y- q)² = r², and that the general equation of a parabola with a vertical axis of symmetry is y = ax² + bx + c. Consider the three points (5, 1), (2, 6), and (1, 3).
1. Find the equation(s) of (all) the circle(s) which pass through the three given points. [4]
2. Find the equation(s) of (all) the parabola(s), if any, with a vertical axis of symmetry which pass through the three given points. [4]
3. In general, three points in a plane that are not all in a straight line determine a unique circle that passes through all three. Explain why this is so using what you know about linear algebra. [2]
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