Hello - I've been stuck on this for a while now and I really need some help.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Find the equations for all lines that are tangent to the circle x^2+y^2=2y and pass through the point (0,4).

2. Relevant equations

y=mx+b

ax^2+bx+c=0

3. The attempt at a solution

From the given equation of the circle, I know that the circle has a center of (0, 1) and that its radius is 1. I also know that the general equation of the tangent line is y-4=m(x-0) or y=m(x-0)+4, which really is simply y=mx+4, right?. I'm trying to solve for the equations of the tangent lines by plugging y=mx+4 into the equation of the circle, x^2+y^2=2y, and solving the resulting quadratic equation. However, I can't seem to solve it. Is this a correct approach to this problem? Here is what I did:

x^2+(mx+4)^2=2(mx+4); simplifying:

x^2+(m^2)(x^2)+6mx+8=0

Assuming this is the right approach so far, I'm not really sure what to do from here. I applied the quadratic formula to the "mx" part of the equation ((m^2)(x^2)+6mx+8) and got -2 and -4, but I think I'm missing something. I'd really appreciate a walk-through.

Many thanks,

John

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# Equation of a tangent line to a given circle from an external point - Stuck!

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