I am trying to solve the following question:
Find the equations of the two circles which pass through the point (2,0) and have both the y-axis and the line y - 1 = 0 as tangents.
This seems like an easy question because you can do it by scale drawing. The first (small) circle can't really go anywhere else - see attachment! But assuming you had to do it by calculation.
To find the equation of a circle we need 3 points on the circumference. We have one supplied directly (2,0) then we have the general equation:
plus the two tangents:
x = 0
y = 1
substituting (2,0) gives:
4 + 4f + c = 0
and substituting x = 0 into the general equation gives:
a quadtratic in y. Since the tangent touches the circle we expect a repeated root i.e.
then substitute y = 1 into the general equation:
again setting gives:
Equations (4) and (5) are correct. What I mean is that when you substitue the correct answers in, they give consistent results but how do you solve them? It seems clear that I am on the wrong track. I then realised that the equation y = -x + 1 must be a diameter to both circles so I tried substituting this into the general equation (1) to solve for the intersection points. It yielded a result for x with many terms in f and g that I couldn't reduce to anything like the correct answer.
Any help here would be appreciated!