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
