Tangents to circle.?

Question

How many common tangents do teo circles x^2+y^2+2x-6y-26=0 & x^2+y^2-4x+2y+4=0 have?

Answer 1

First find the centers and radii of the circles.

First circle.

x² + y² + 2x - 6y - 26 = 0
(x² + 2x) + (y² - 6y) = 26
(x² + 2x + 1) + (y² - 6y + 9) = 26 + 1 + 9
(x + 1)² + (y - 3)² = 36

Center is (-1, 3). Radius is 6.
__________

x² + y² - 4x + 2y + 4 = 0
(x² - 4x) + (y² + 2y) = -4
(x² - 4x + 4) + (y² + 2y + 1) = -4 + 4 + 1
(x - 2)² + (y + 1)² = 1

Center is (2, -1). Radius is 1.
___________

The distance between the centers is the distance between
(-1, 3) and (2, -1).

d = √[(-1 - 2)² + (3 + 1)²] = √(3² + 4²) = √(9 + 16)
d = √25 = 5

Since the centers are a distance 5 apart and the radius of the larger circle is 6, the center of the smaller circle is inside the larger one. The distance of farthest point on the smaller circle from the center of the larger circle is

5 + 1 = 6

This is the same as the radius of the larger circle. So the two circles are internally tangent. The number of common tangent lines to the two circles is therefore one.