# Equation of a Circle with a Center and Tangent Point

1. Jul 30, 2012

### xxmegxx

What is the equation of the circle with a center point of (10, -14) when the circle is tangent to x=13?

D = √(13-10)^2 + (0-(14))^2
D = √(3)^2 + (14))^2
D = √9+196
D = √205

(x-10)^2 + (y-(-14))^2 = √205^2
(x-10)^2 + (y+14)^2 = 205

But how am I suppose to graph this?

Last edited: Jul 30, 2012
2. Jul 30, 2012

### Villyer

3. Jul 30, 2012

### Curious3141

Your method of calculating the radius (if D is supposed to be the radius) makes no sense.

The problem is actually very simple. You're given that the circle is tangent to x=13, which is a vertical line. You know the centre has an x-coordinate of 10. So what can you say about the radius?

4. Jul 30, 2012

### xxmegxx

I meant D to be the distance. I'm not sure how you solve this problem without graph paper.

5. Jul 30, 2012

### Curious3141

Distance from what to what?

There's no need for graph paper. All you need is a reasonable sketch. Remember the general equation for the circle and what the terms represent.

6. Jul 30, 2012

### xxmegxx

The distance from the center point to the tangent line to find the radius.

7. Jul 30, 2012

### Integral

Staff Emeritus
What is the issue with graphing? it is a circle, you know the center and radius. What else do you want?

8. Jul 30, 2012

### Curious3141

But that distance is NOT the radius! Remember that x=13 is a vertical tangent to the circle. A circle with the radius you calculated would not have that line as a tangent.

Also, what you calculated was the distance between the points (10,-14) and (13,0). This is NOT the same as the (shortest) distance between (10,-14) and the line x = 13. Do you see why?

Just do a sketch. Do you see why the radius is simply 13 - 10 = 3?

(As a final point, there was an error in your working in the first post. The distance should've been $[(13 - 10)^2 + (0 - (-14))^2]^\frac{1}{2}$. Note the sign in the y-term. But the squaring masked your error.)

Last edited: Jul 30, 2012
9. Jul 31, 2012

### HallsofIvy

Staff Emeritus
Just to avoid confusion, what he is writing here is the radius. But you are saying that the distance calculated before, from the center of the circle to the point (13, 0), is not "the distance from the center point to the tangent line".

I suspect that the real difficulty is that xxmeqxx is thinking, incorrectly, that "x= 13" means the point on the x-axis with x-component 13 rather than, as every here is telling him, the line of all points whose x-component is 13, (13, y).

Last edited: Jul 31, 2012
10. Jul 31, 2012

### Curious3141

Yes, what he wrote in words (with reference to distance between centre and tangent line) is the radius. But what he calculated (distance between centre and (13,0) ) is not. That's what I meant.