Given circle, find the line of bearing of tan lines thru 0,0

In summary, the conversation discusses finding a general equation in terms of x, y, R, and Θ to determine the angle each tangent line makes with the x axis. The solution involves using trigonometry and considering that the tangent line will be perpendicular to the radius where it intersects the circle. The concept is further explained with the use of diagrams and a possible theorem is mentioned. A correction is also made to the diagram in the original post.
  • #1
srcs
4
1
This question occurred to me a few days ago and it's been bugging me ever since.

Consider a circle in the coordinate plane with center P(x,y) and radius R, where R < D, D being the distance from the origin to the circle's center.

There are two lines tangent to the circle (T1 and T2) that pass through the origin.

The circle's location can also be considered as a line segment of radius D and some angle Θ with the x axis.

How can I find a general equation in terms of x, y, R, and Θ to determine the angle each tangent line makes with the x axis?

073G7O0.png


Solution found:

C9qMSEA.png


Adding Θs to Θdx should yield the bearing for the other tangent line.
 
Last edited:
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  • #2
You might consider that where the line hits the circle it will be perpendicular to some radius so you have some D length and some angle that D makes with the X axis and some trig to mix into get your answer.
 
  • #3
jedishrfu said:
You might consider that where the line hits the circle it will be perpendicular to some radius so you have some D length and some angle that D makes with the X axis and some trig to mix into get your answer.

Solved. The help is appreciated.
 
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  • #4
I can't tell if this is just a coincidence of the values I chose, but the distance from the x-axis to the circle's center (Dy) worked out to be equal to the distance from the origin to the tangent line's intersection (T). Weird.
 
  • #5
srcs said:
I can't tell if this is just a coincidence of the values I chose, but the distance from the x-axis to the circle's center (Dy) worked out to be equal to the distance from the origin to the tangent line's intersection (T). Weird.

This may be a theorem and might be proven if you think about it more abstractly. You are actually saying that drawing a circle with center at the origin that passes through the original circle's center will intersect at the tangent line points.

Perhaps @Mark44 could verify your observation and point to the theorem if it exists as I don't know for sure.

Edit: okay I think this is it on Wikipedia

https://en.m.wikipedia.org/wiki/Tangent_lines_to_circles

Look down a bit in the article to see two intersecting circles and how tangent lines from one circle go through the center of another.
 
  • #6
jedishrfu said:
This may be a theorem and might be proven if you think about it more abstractly. You are actually saying that drawing a circle with center at the origin that passes through the original circle's center will intersect at the tangent line points.

Perhaps @Mark44 could verify your observation and point to the theorem if it exists as I don't know for sure.

Edit: okay I think this is it on Wikipedia

https://en.m.wikipedia.org/wiki/Tangent_lines_to_circles

Look down a bit in the article to see two intersecting circles and how tangent lines from one circle go through the center of another.
Thanks, this is interesting. Also I just noticed an error in my second diagram when I renamed my variables: Θty should beΘtx everywhere except in the cutout triangle, where it's correctly labeled. I'm pretty sure the calculations themselves are still correct. I'll update the image.

Never mind, the OP no longer has an edit option. I'll just post it here.

WIw8iv5.png
 
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Likes jedishrfu

1. What is the definition of a circle?

A circle is a closed shape in a plane that is formed by tracing a point that moves in a plane such that the distance from the point to a fixed point, called the center, remains constant.

2. How can the line of bearing of tan lines through 0,0 be found?

The line of bearing of tan lines through 0,0 can be found by first finding the slope of the tangent line at the point (0,0) using the formula y = mx, where m is the slope. Then, the angle of the tangent line can be determined using inverse trigonometric functions.

3. What is the significance of finding the line of bearing of tan lines through 0,0?

Finding the line of bearing of tan lines through 0,0 is important in understanding the direction and orientation of the circle. It can also be used in various mathematical and scientific applications, such as in navigation and engineering.

4. Can the line of bearing of tan lines through 0,0 be negative?

Yes, the line of bearing of tan lines through 0,0 can be negative. This indicates that the tangent line is sloping downwards as it approaches the point (0,0).

5. Are there any other methods for finding the line of bearing of tan lines through 0,0?

Yes, there are other methods for finding the line of bearing of tan lines through 0,0. One method is using the derivative of the circle equation, which is the same as the slope of the tangent line. Another method is using trigonometric identities to calculate the angle of the tangent line.

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