First, we start with the standard circle equation. (x-h)2+(y-k)2=r2
Then, we can plug in the center coordinates we're given to this standard equation. So when we do that we get this: (x-1)2+(y--3)2=r2. We can simplify that to (x-1)2+(y+3)2=r2.
So what's missing from our standard equation is the r. How do we find that? We graph the things we know: the center of the circle (1, -3) and the tangent line x=6.
When you graph, you have that point at (1, -3) and then you should have a vertical line that crosses the x-axis at (6,0).
Since we know the center of our circle is (1, -3) and we know that at any tangent line to a circle, the distance from the center to that tangent line intersecting the circle is the radius. If you want to visually see it, start from (6,0) and draw a circle so that (1, -3) is in the center.
But anyways, we know the distance from the center to the tangent line is the radius. So now we just need the distance. So follow the vertical line you drew until you're at (6, -3) and find the distance from that to (1, -3) and it should be 5. Therefore, r=5.
Now we can finish the circle equation. (x-1)2+(y+3)2=52 or simplified (x-1)2+(y+3)2=25
