Question

find the coordinates of the centre of the circle passing through the points A (-2,-3),B(-1,0) and C(7 -6). Also find the radius of this circle

Answer 1

Answer:

The first thing to note is that if you specify two lines and a point, then there are two circles that pass through the point and are tangent to the lines. We can confirm this with a little math:

Let the lines be x+utx+ut and x+vtx+vt where xx is a general point, uu and vv are unit vectors and tt is a real number, so that the lines cross at xx (the case where the lines are parallel can be treated separately). Then the line on which the centre of a circle which is tangent to both lines can lie is given by

x+u+v2t

x+u+v2t

We also require that the circle goes through the point yy, so we need the tt such that

||x+u+v2t−y||2=r(t)2

||x+u+v2t−y||2=r(t)2

where r(t)r(t) is the radius of the circle, given by r=ttan(θ/2)r=ttan⁡(θ/2), where θθ is found from cosθ=u⋅vcos⁡θ=u⋅v (draw a diagram and you'll see where this expression comes from). Expanding this out, we get

||x−y||2−(u+v)⋅(x−y)t+14||u+v||2t2=t2tan2(θ/2)

||x−y||2−(u+v)⋅(x−y)t+14||u+v||2t2=t2tan2⁡(θ/2)

which is a quadratic equation for tt. Solving it gives the the values of tt corresponding to the two possible circles