Hello

If I have these two circles:

(x + 1)^2 + y^2 = 25

and

(x - 2)^2 + (y-1)^2 = 9

How do I work out where the circles (or if) intersect?

It is difficult to set eg y= due to the different terms of x.

Angus

The set of all points a distance of 5 units from (-1,0) is given by the equation

\(\displaystyle [x-(-1)]^2+[y-0]^2=5^2\Rightarrow\ (x+1)^2+y^2=25\Rightarrow\ x^2+y^2+2x-24=0\)

The set of all points a distance of 3 units from (2,1) is given by

\(\displaystyle (x-2)^2+(y-1)^2=3^2\Rightarrow\ x^2-4x+4+y^2-2y+1=9\Rightarrow\ x^2+y^2-4x-2y-4=0\)

To find the points (x,y) that are both 5 units from (-1,0) and 3 units from (2,1)

then the "x" values are equal in both equations.

The "y" values are also equal.

0-0=0

\(\displaystyle \left[x^2+y^2+2x-24\right]-\left[x^2+y^2-4x-2y-4\right]=x^2-x^2+y^2-y^2+2x+4x+2y-24+4=0\)

\(\displaystyle 6x+2y-20=0\Rightarrow\ 3x+y-10=0\)

However, the resulting equation is the set of all points equidistant from the centres.

This is because we obtain the same result by subtracting the equations of

other circles with these centres, but whose radii squared differ by the exact same amount.

If you wish, you can then find the point of intersection of this line with either circle, by again allowing the "x" values to be equal and the "y" values to also be equal.