If the circles x^2 + y^2 + 12x + 6y - 19 = 0 and x^2 + y^2 - 4x - 6y = k are tangent to each other, find the possible values of k.
How can I know whether the unknown circle is inside or outside the circle x^2 + y^2 + 12x + 6y - 19 = 0?
or without knowing this, can I still solve the problem?
Ah... sorry that I didn't describe my question well......
Let C1 be the circle x^2 + y^2 + 12x + 6y - 19 = 0
and C2 be the circle x^2 + y^2 - 4x - 6y = k .
From the equations, we can find that:
Centre of C1 = G1 = (-6 , -3)
radius of C1 = r1 = 5
Centre of C2 = G2 = (2 , 3)
radius of C2 = r2 = (13 + k) ^(1/2)
G1G2 = 10
If the circles are tangent to each other,
Case (I) C1 is out of C2
r1 + r2 = G1G2
Case (II) C1 is in C2
Difference of radii(r1 and r2) = G1G2
I can find one of the answers by consider Case (I).
But I don't understand why I can find all the answer by considering case (II) with taking the value of 'Difference of radii(r1 and r2) ' as |r2-r1|.
The answers are -9 and 311.
Can I always apply absolute value to solve these type of problems?
I know for Case (II),
the radius of required circle must be greater than the given circle,
i.e. r2-r1 = G1G2.
and this gives an answer k = 331 ...
But I don't understand why if I take |r2 - r1| = G1G2,
I can find the answer k = 331 and also the the answer k = -9, which is the one I found by considering Case(I). Is this just a coincidence? Or I can do like this every time?
As you can see in the diagram above, the two circles, the blue and the magenta, are centered at (2,3) and both are tangent to the red circle centered at (-6,-3) with radius 5. The two values of k are k=5 gives the blue circle and k=212 gives the magenta circle.