Question 24545
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^2 + y^2 - 6x + 4 = 0}}}
{{{(x^2 -6x) + (y^2) = -4}}}
{{{(x^2 -6x + 9) + (y^2 + 0) = -4 + 9 + 0}}}
{{{(x-3)^2 + (y+0)^2 = 5}}}


so, centre is (3,0) and radius is {{{sqrt(5)}}}.


{{{x^2 + y^2 - 2x + 8y + 12 = 0}}}
{{{(x^2 - 2x) + (y^2 + 8y) = -12}}}
{{{(x^2 - 2x + 1) + (y^2 + 8y + 16) = -12 + 1 + 16}}}
{{{(x-1)^2 + (y+4)^4 = 5}}}


so centre is (1,-4) and radius is {{{sqrt(5)}}}.


So, now for the proof:

Distance between 2 centres is found using Coordinate Geometry:

distance = {{{sqrt((3-1)^2+(0--4)^2)}}}
distance = {{{sqrt((3-1)^2+(0+4)^2)}}}
distance = {{{sqrt((2)^2+(4)^2)}}}
distance = {{{sqrt(4+16)}}}
distance = {{{sqrt(20)}}}
distance = {{{sqrt(4*5)}}}
distance = {{{sqrt(4)sqrt(5)}}}
distance = {{{2sqrt(5)}}}


And radii added together are {{{sqrt(5) + sqrt(5)}}} which is also {{{2sqrt(5)}}}... so proved :-)


jon.