SOLUTION: show analytically that the circles x(squared) + y(squared) - 6x + 4=0 and x(squared) + y(squared) - 2x + 8y + 12=0 are tangent to each other.

Algebra ->  Circles -> SOLUTION: show analytically that the circles x(squared) + y(squared) - 6x + 4=0 and x(squared) + y(squared) - 2x + 8y + 12=0 are tangent to each other.       Log On


   

Question 24545: show analytically that the circles x(squared) + y(squared) - 6x + 4=0 and x(squared) + y(squared) - 2x + 8y + 12=0 are tangent to each other.
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
well, if the 2 circles are tangents, it means they just touch. This would mean that the length of the line joining their centres will equal their 2 radii added.

So, that is the plan...find the centres and radii lengths and then prove that the length between the 2 centres is equal to the radii added together.
x%5E2+%2B+y%5E2+-+6x+%2B+4+=+0
%28x%5E2+-6x%29+%2B+%28y%5E2%29+=+-4
%28x%5E2+-6x+%2B+9%29+%2B+%28y%5E2+%2B+0%29+=+-4+%2B+9+%2B+0
%28x-3%29%5E2+%2B+%28y%2B0%29%5E2+=+5

so, centre is (3,0) and radius is sqrt%285%29.

x%5E2+%2B+y%5E2+-+2x+%2B+8y+%2B+12+=+0
%28x%5E2+-+2x%29+%2B+%28y%5E2+%2B+8y%29+=+-12
%28x%5E2+-+2x+%2B+1%29+%2B+%28y%5E2+%2B+8y+%2B+16%29+=+-12+%2B+1+%2B+16
%28x-1%29%5E2+%2B+%28y%2B4%29%5E4+=+5

so centre is (1,-4) and radius is sqrt%285%29.

So, now for the proof:
Distance between 2 centres is found using Coordinate Geometry:
distance = sqrt%28%283-1%29%5E2%2B%280--4%29%5E2%29
distance = sqrt%28%283-1%29%5E2%2B%280%2B4%29%5E2%29
distance = sqrt%28%282%29%5E2%2B%284%29%5E2%29
distance = sqrt%284%2B16%29
distance = sqrt%2820%29
distance = sqrt%284%2A5%29
distance = sqrt%284%29sqrt%285%29
distance = 2sqrt%285%29

And radii added together are sqrt%285%29+%2B+sqrt%285%29 which is also 2sqrt%285%29... so proved :-)

jon.