__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 x^{2}
+ y^{2} = r^{2}

Hence in this^{ }case r^{2}
is 9 so the equation of the circle must be x^{2} + y^{2} = 9 since x^{2}
+ y^{2 }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
x^{2} + y^{2} = 9

Putting 1 into 2...
x^{2} + (2x)^{2} = 9

x^{2} + 4x^{2} = 9

5x^{2} = 9

x^{2} = 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.