Question

Write the equation of the circle with center (-2,-3) and tangent to the line 2x+5y=10.

Answer 1

let P(a,b) be the point on the circle where the tangent hits it.
slope of the given tangent is -2/5
so the slope of the radius to P is +5/2

but slope of radius is (b+3)/(a+2)
so (b+3)/(a+2) = 5/2
5a - 2a = -4 (#1)

but P(a,b) also lies on the tangent, so
2a + 5b = 10 (#2)

taking 5(#1) + 2(#2) and adding gave me a=0
subbing that back into #2 gave be b=2

So the point of tangency is (0,2)
which gives me a radius of √29 using the distance formula between P and the centre.

so now you have the centre and the radius.....
(x+2)^2 + (y+3)^2 = 29