# Thread: Tangent Line of a Circle

1. ## Tangent Line of a Circle

Hey guys, I've tried this one a few times and just can't seem to get it. Help would be appreciated:

A circle has its center at C(4,8). A point Q(12,6) is on the circumference of the circle. A tangent line to the circle is drawn at point Q. Determine the equation of the tangent line.

2. Originally Posted by Vesicant
Hey guys, I've tried this one a few times and just can't seem to get it. Help would be appreciated:

A circle has its center at C(4,8). A point Q(12,6) is on the circumference of the circle. A tangent line to the circle is drawn at point Q. Determine the equation of the tangent line.
Since you originally posted this in the Pre-algebra and Algebra subforum (and I moved it to here) I assume that you're to do this without using calculus ..... Correct?

3. Originally Posted by mr fantastic
Since you originally posted this in the Pre-algebra and Algebra subforum (and I moved it to here) I assume that you're to do this without using calculus ..... Correct?
Correct

4. Originally Posted by Vesicant
Correct
Draw a diagram. Note the fact that the angle between the tangent and the radius is 90 degrees. Use this fact, geometry and some simple trigonometry to get the angle $\theta$ that the tangent makes to the x-axis. Then $m = \tan \theta$.

Knowing the gradient of a line and a point on it, you should be able to get the equation of the line.

5. You might also use the fact that if the lines corresponding to $y= m_1x+ b_1$ and $y= m_2 x+ b_2$ are perpendicular, then $m_1m_2= -1$. You know that a radius passing through (4, 8) and (12, 6). What is the slope of that line? You know that the tangent at (12, 6) is perpendicular to that radius so you can find the slope of the tangent. Now you know the slope and one point of the line so you can find the equation of the line.

6. Originally Posted by HallsofIvy
You might also use the fact that if the lines corresponding to $y= m_1x+ b_1$ and $y= m_2 x+ b_2$ are perpendicular, then $m_1m_2= -1$. You know that a radius passing through (4, 8) and (12, 6). What is the slope of that line? You know that the tangent at (12, 6) is perpendicular to that radius so you can find the slope of the tangent. Now you know the slope and one point of the line so you can find the equation of the line.
So if the slope given by (4,8) and (12,6) is $M=8-6/4-12=2/-8=-1/4$ then the slope of the tangent is $4/1$ which would make the equation of the tangent $Y=4/1x+3$?

Here's a quick graph I did of it, Something doesn't feel right though :/

7. How did you get "+3"? Yes, the slope is 4 so the equation is of the form y= 4x+ b.
In order that it pass through (12, 6) you must have 6= 4(12)+ b so b= 6- 48= -42.
Try y= 4x- 42.

8. Originally Posted by HallsofIvy
How did you get "+3"? Yes, the slope is 4 so the equation is of the form y= 4x+ b.
In order that it pass through (12, 6) you must have 6= 4(12)+ b so b= 6- 48= -42.
Try y= 4x- 42.
Yeah I know what I did wrong lol, Thought about that while sitting in class today for some reason I put the X intercept into the equation haha.

So yes $$Y=4x-42$$ is what I got after reworking it, Thanks for all the help.