show that the circles are tangent to each other x^2+y^2-6x+1=0 x^2+y^2-2y+8x-1=0.analytic geometry

The circles are tangent to each other if they touch each
other only at a single point.

We have the equation of the
circles:

x^2 + y^2 - 6x + 1 = 0
...(1)

x^2 + y^2 - 2y + 8x - 1 = 0
...(2)

Equating (1) and
(2)

x^2 + y^2 - 6x + 1 = x^2 + y^2 - 2y + 8x -
1

=> - 6x + 1 + 2y - 8x + 1 =
0

=> -14x + 2y + 2 =
0

=> -7x + y + 1 =
0

=> y = -1 +
7x

substitute this in (1)

x^2
+ (-1 + 7x)^2 - 6x + 1 = 0

=> x^2 + 1 + 49x^2 - 14x
- 6x + 1 = 0

=> 50x^2 - 20x + 2 =
0

=> 50x^2 - 10x - 10x +2
=0

=> 10x(5x - 1) - 2(5x - 1) =
0

=> (10x - 2)(5x - 1) =
0

=> x = 2/10 = 1/5 and x =
1/5

y = -1 + 7x = -1 + 7/5 =
2/5

The two circles meet only at the point (1/5 ,
2/5)

This proves that they are tangent to
each other