 # Linear Algebra For Quadratic Curves

## Question

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. 
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. 
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. 

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