Find a line through point given a distance to another point

Question

Find the line(s) through (-1, 4) for which distance to (6, 3) is 5.

Since it is a homework question, please give some directions on how to tackle it first, not the whole solution.

Answer 1

Let the line be $(y-4)=k(x+1)$, or,

$$kx-y+k+4=0$$

Use the point-line distance formula

$$\frac{|6k-3+k+4|}{\sqrt{1+k^2}}=5$$

Simplify

$$12k^2+7k-12=0$$

Solve for the slope $k$,

$$k_1=-\frac 43,\>\>\>\>\>k_2=\frac34$$

Then, plug the slopes into the line equations above. Note that they are two lines satisfying the given.

Answer 2

What are the points for what you have distance $5$ of $(6,3)$?

Remember that the tangent to a circle has angle $90^\circ$ with the radius.