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Equations of Circles

This is a circle, radius 3cm

Triangles can be drawn within it where the radius is the hypotenuse

Pythagoras' Theorem says that x2 + y2 = r2

Hence in this case r2 is 9 so the equation of the circle must be x2 + y2 = 9 since x2 + y2 always needs to equal the square of the hypotenuse.

You will be asked what the formulas of circles centred on the origin are and to find the coordinates of where a line cuts the circle using simultaneous equations.




For example; where does the line y=2x cross the circle?

Simultaneous equation 1    y=2x

Simultaneous equation 2    x2 + y2 = 9

Putting 1 into 2...               x2 + (2x)2 = 9

                                        x2 + 4x2 = 9

                                        5x2 = 9

                                        x2 = 1.8

                                        x = +√2.8 or -√2.8     hence y=+2√2.8 or -2√2.8

*Notice how the lines cross at both (+√2.8 , +2√2.8) and (-√2.8 , -2√2.8). The reason there are two solutions is because the line crosses the circle twice! Try Drawing it to scale to see that this solution makes sense*

NB) If you only get one solution for a question like this, it means that the line and the circle only share one coordinate, not two, so your line must be a tangent.


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