(c) show that the circles x2 + y2 −16x−12y + 75 = 0 and 5x2 + 5y2 −32x−24y + 75 = 0 touch each other.

so, let's do the first one



now, in a perfect square trinomial, we know the middle term is just the product of 2 times the square root of the term on the left end, times the square root of the term on the right

that means 

so, those are our missing values, now
bear in mind all we're doing is borrowing from our good friend Mr Zero, 0
so, if we add 8² and 6², we also have to subtract 8² and 6²

then  

so, that's the equation of the circle for the first equation, centered at 8,6 and with a radius of 5

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now, onto the 2nd equation, we'll do the same







so, that's a circle centered at 16/5 and 12/5, with  a radius of 1
so .those are the two circle's equations

notice, the picture below, the radius of 5, the first equation, is the bigger circle

It’s hard to understand what you put.. can you show a picture ?