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
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.