finding the length of a tangent?

Question

1)find the lengths of the tangents from point (3,4) to the circle x^2 +y^2 = 4

i think the answer is sqrt 21 not sure on working though


2) find the lengths of the tangents from point (5,7) to the circle x^2 + y^2 - 2x - 4y - 4 = 0

i think the answer is 4 sqrt 2 not sure on working again though

Answer 1

The radius is 2 and the distance of(3,4)to the center(0,0) =5
so length = sqrt(5^2-2^2)=sqrt(21)
Remember the tangent is perpendicular to the radius passing through the contact point
2) the center is (1,2) and the radius r^2= 9
The distance between (1,2) and (5,7) d^2=25+16=41
so length= sqrt(41-9) = sqrt(32) = 4*sqrt(2)

Answer 2

I'll do the first one:
Let the tangent intersect the circle at a point
(p, +sqrt(4-p^2))

The slope of this line is the same as the slope of the circle at that point.
x^2 + y^2 = 4
2x + 2y*dy/dx = 0
dy/dx = -x/y
= -p / sqrt(4-p^2) ...(1)

Also slope = (4 - sqrt(4-p^2)) / (3-p) ...(2)

Now you need to solve the equation:
-p / sqrt(4-p^2) = (4 - sqrt(4-p^2)) / (3-p)

You'll get
p = [12 - 8*sqrt(21)] / 25

Now find the difference between this point and (3,4)
That will give you sqrt(21).

*EDIT*
You'll like the geometry approach by santmann much better than this calculus approach. I only used this method because I recently got the software Maple, and I'm testing it out.