Tangents on a circle

The general form of a circle is $$(x-h)^2 + (y-k)^2 = r^2$$. Where (h,k) is the centre of the circle.
Another general form is $$x^2 + y^2 + 2gx + 2fy + c = 0$$.  The centre of the circle is (-g, -f) and the radius is $$\sqrt{\ g^2+f^2-c}$$
Comparing with our equation $$x^2+y^2-12\sqrt2 x-12 \sqrt2y+95 = 0. ==> g= -6 \sqrt2 and f= -6 \sqrt2$$
Therefore centre is $$(-6 \sqrt2 ,6 \sqrt2)$$
Substituting this in the first general form of the circle, we get

The first circle is $$x^2+y^2 = 25$$ and the second circle is $$(x-6\sqrt2 )^2 + (y-6\sqrt2 )^2 = 49$$.
The centers of the two circles are therefore $$(0,0)$$ and $$(6\sqrt{2},6\sqrt{2})$$
The distance between the centers is 12, and the sum of the radii is 12.
The circles touch each other at one point, and their centers lie on opposite sides of the point of contact.
The number of common tangents is, therefore, 3.

The given two equations are the equations of two circles. We need to calculate how many common tangents both the circles have. So, we need to first find out the centres of the two circles and the radii of the two circles. Once we have those two, we will understand at how many points the circles intersect each other. And from then we can calculate the number of tangents the circles have in common. 

The radius and centre of the circles is as follows => $$x^2 + y^2 + 16x - 10y - 32=0$$

So, $$(x^2 + 16x + 64) + (y^2-10y+25) = 121$$

So, $$ (x+8)^2 + (y-5)^2 = 11^2 $$.

Therefore, Centre (-8,5) and radius 11.

$$ x^2 + y^2 - 8x + 12 = 0$$

So, $$ (x^2 - 8x +16) + y^2 = 2^2$$

So, $$ (x-4)^2 + y^2 = 2^2$$.

Therefore, centre (4, 0) and radius = 2.

The distance between the two centres = $$\sqrt{ (-8-4)^2 + (5-0)^2} = \sqrt{ 12^2+5^2} = 13$$.

Sum of radii = 11+2 = 13.

Since distance between 2 centres is equal to the sum of radii

Hence the two circles touch each other at one point. Hence there are 3 common tangents, all of which are direct common  tangents

The distance between the circles is 14 cm, which is greater than the sum of the radii (7+5 = 12). So, the circles have no point of intersection between them. The number of common tangents between the two circles is, therefore, 4.