How Do You Find the Equation of a Tangent Line to a Circle?

Click For Summary

Homework Help Overview

The problem involves finding the equation of a tangent line to a circle, given the center of the circle at (3,2) and the point of tangency at (8,4). Participants are exploring the relationship between the radius and the tangent line, as well as the necessary equations involved.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the radius of the circle and its calculation, questioning whether it is 5 or sqrt(29). There is exploration of the slope of the radius and its relationship to the slope of the tangent line. Some participants suggest using the point-slope form of a line to find the tangent line's equation.

Discussion Status

There is ongoing exploration of the slopes involved, with some participants confirming the slope of the radius as 2/5 and discussing the implications for the slope of the tangent line. Guidance has been offered regarding the relationship between the slopes of the radius and the tangent line, but no consensus has been reached on the final equation of the tangent line.

Contextual Notes

Participants note that the question asks specifically for the equation of the tangent line and the equation of the circle in polynomial form, but some express confusion about the relevance of the circle's equation to finding the tangent line.

nukeman
Messages
651
Reaction score
0

Homework Statement



Center of circle is: (3,2)
Tangent point: (8,4)

Question: What is the equation of the tangent line?


Homework Equations





The Attempt at a Solution



I am just not getting it.

So, would the radius be 5?

Now, would i go:

(x-3)^2 + (y-2)^2 = 5

? Any help would be great!
 
Physics news on Phys.org
hi nukeman! :smile:

the tangent will be perpendicular to the radius :wink:
 
tiny-tim said:
hi nukeman! :smile:

the tangent will be perpendicular to the radius :wink:

ooook. So, where I do from there?

The line would have a point of (8,4), and x intercept would be (9,0)

would the slope be 2/5 ?

Man, I am lost
 
nukeman said:
… x intercept would be (9,0)

where do you get that from? :confused:

what is the slope of the radius? :smile:​
 
Would the slope be 2/5 ?
 
nukeman said:
Would the slope be 2/5 ?

yup! :smile:

so the slope of the tangent is … ? :wink:​
 
nukeman said:

Homework Statement



Center of circle is: (3,2)
Tangent point: (8,4)

Question: What is the equation of the tangent line?


Homework Equations





The Attempt at a Solution



I am just not getting it.

So, would the radius be 5?

Now, would i go:

(x-3)^2 + (y-2)^2 = 5

? Any help would be great!

This problem can be done without having to find the equation of the circle or its radius, but to set the record straight, the radius of the circle is not 5. The formula for the distance between two points (x1, y1) and (x1, y1), is sqrt((x2 - x1)2 + (y2 - y1)2). Applying this formula gives you sqrt(29) for the radius of the circle.

The equation of this circle would then be (x-3)2 + (y-2)2 = (sqrt(29))2 = 29.
 
Oh ok, thanks Mark!

So, that would give me the equation of the circle. How would I get the equation of the tangent line?
 
nukeman said:
Oh ok, thanks Mark!

So, that would give me the equation of the circle. How would I get the equation of the tangent line?

what is the slope of the radius out to the point where the tangent line hits?

what's the realtionship between the slope of the radius and the slope of the tangent line?

what is one point on the tangent line?
 
  • #10
As I said, the equation of the circle doesn't really enter into the problem.

If you know the slope of a line and a point on it, you can use the point-slope form of the equation of a line, which is y - y0 = m(x - y0).

Once you get the slope of the radius between the circle center and the point (8, 4), you can find the slope of the tangent line. The tangent line and the radius will be perpendicular, so what does that say about their slopes?
 
  • #11
So, they would have the same slope then?

The question in my book asks for

*The Equation of the Tangent Line
*The Equation of the Circle, stated in polynomic form
 
  • #12
nukeman said:
So, they would have the same slope then?

The question in my book asks for

*The Equation of the Tangent Line
*The Equation of the Circle, stated in polynomic form

Nukeman, I do not mean any disrespect here, but if getting the equation of the line is not completely clear to you based on what everyone has been telling you, you really need to study the basics a bit more thoroughly.
 
  • #13
I got how the equation of the circle would be (x-3)2 + (y-2)2 = (sqrt(29))2 = 29

The answer I got the equation to the tangent line comes out to, in

y-4 = 2/5x - 8
y= 2/5x - 2
 
  • #14
nukeman said:
I got how the equation of the circle would be (x-3)2 + (y-2)2 = (sqrt(29))2 = 29
This isn't relevant to the problem.
nukeman said:
The answer I got the equation to the tangent line comes out to, in

y-4 = 2/5x - 8
y= 2/5x - 2
Nope, you're missing an important point. The slope of the radius between (3, 2) and (8, 4) - the point of tangency - is 2/5, but that isn't the slope of the tangent line. The tangent line is perpendicular to the radius, so the slope of the tangent line will be ?.
 
  • #15
But this is the equation to the circle correct? (x-3)2 + (y-2)2 = (sqrt(29))2 = 29

Ok...Is the slope then, -5/2 (minus 5 over 2) ?
 
  • #16
nukeman said:
But this is the equation to the circle correct? (x-3)2 + (y-2)2 = (sqrt(29))2 = 29
Yes, but so what? The question isn't asking for the equation of the circle.
nukeman said:
Ok...Is the slope then, -5/2 (minus 5 over 2) ?
YES

Now, find the equation of the tangent line. Forget the equation of the circle.
 
  • #17
so, its

y-4 = -5/2 (x-8) ?

or,

y= -5/2x -1/2

?
 
  • #18
Last edited by a moderator:

Similar threads

Family of lines that are at a distance of 5 from the origin
  • · Replies 14 ·
Replies
14
Views
1K
Find the center of a circle given a tangent line & point
  • · Replies 2 ·
Replies
2
Views
4K
What is the equation of the tangent line to a circle passing through the origin?
  • · Replies 5 ·
Replies
5
Views
2K
High School Circle tangent to two lines and another circle
  • · Replies 2 ·
Replies
2
Views
6K
Finding Tangent Equation from (-5,4) to Circle
  • · Replies 10 ·
Replies
10
Views
3K
How Do You Calculate the Length of TP in a Common Tangents Problem?
  • · Replies 4 ·
Replies
4
Views
2K
Circle Tangents and Locus Problem: Finding the Equation with Given Slopes
  • · Replies 20 ·
Replies
20
Views
3K
How Do Three Tangents Form an Equilateral Triangle Around a Circle?
  • · Replies 7 ·
Replies
7
Views
3K
Curve and tangent line problem, finding the area of enclosed region
  • · Replies 3 ·
Replies
3
Views
3K
Equations of a line tangent to a circle
  • · Replies 3 ·
Replies
3
Views
4K