- #1
MarkFL
Gold Member
MHB
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Here is the question:
I have posted a link there to this topic so the OP can see my work.
I have posted a link there to this topic so the OP can see my work.
Find Where Two Tangent Lines Intersect on a Circle | Math Help
In summary, to find where two tangent lines intersect on a circle, you can simplify the problem by orienting the circle's center at the origin and rotating it so that the tangent points have the same x-coordinate. Then, using the distance formula and the equation of the circle, you can find the value of d, which represents the distance from the origin to the point where the tangent lines intersect. This process can be used to solve for the intersection point of any two tangent lines on a circle.
- #2
MarkFL
Gold Member
MHB
- 13,288
- 12
Hello Getty,
We can greatly simplify this problem, if we orient the circle's center at the origin of our coordinate system, and rotate the circle such that the two tangent points have the same $x$-coordinate (where $0<x<r$), one point in the first quadrant, and one in the fourth quadrant.
Please refer to the following diagram:
View attachment 1164
Because $r$ and $\ell$ are perpendicular, we may state:
\(\displaystyle r^2+\ell^2=d^2\)
Using the distance formula, we find:
\(\displaystyle \ell^2=(x-d)^2+y^2\)
and from the equation of the circle, we have:
\(\displaystyle y^2=r^2-x^2\)
Hence, we may now write:
\(\displaystyle r^2+(x-d)^2+r^2-x^2=d^2\)
\(\displaystyle 2r^2+x^2-2xd+d^2-x^2=d^2\)
\(\displaystyle r^2-xd=0\)
\(\displaystyle d=\frac{r^2}{x}\)
We can greatly simplify this problem, if we orient the circle's center at the origin of our coordinate system, and rotate the circle such that the two tangent points have the same $x$-coordinate (where $0<x<r$), one point in the first quadrant, and one in the fourth quadrant.
Please refer to the following diagram:
View attachment 1164
Because $r$ and $\ell$ are perpendicular, we may state:
\(\displaystyle r^2+\ell^2=d^2\)
Using the distance formula, we find:
\(\displaystyle \ell^2=(x-d)^2+y^2\)
and from the equation of the circle, we have:
\(\displaystyle y^2=r^2-x^2\)
Hence, we may now write:
\(\displaystyle r^2+(x-d)^2+r^2-x^2=d^2\)
\(\displaystyle 2r^2+x^2-2xd+d^2-x^2=d^2\)
\(\displaystyle r^2-xd=0\)
\(\displaystyle d=\frac{r^2}{x}\)
Attachments
Related to Find Where Two Tangent Lines Intersect on a Circle | Math Help
1. How do you find the point of intersection between two tangent lines on a circle?
To find the point of intersection between two tangent lines on a circle, you can use the properties of tangents and circles. First, find the slope of each tangent line at the point of tangency by finding the derivative of the circle's equation. Then, use the point-slope form of a line to create equations for each tangent line. Finally, solve the system of equations to find the point of intersection.
2. Can there be more than one point of intersection between two tangent lines on a circle?
No, there can only be one point of intersection between two tangent lines on a circle. This is because a tangent line only touches a circle at one point and does not intersect it at any other point.
3. How does the radius of the circle affect the point of intersection between two tangent lines?
The radius of the circle does not affect the point of intersection between two tangent lines. This is because the point of tangency, where the tangent lines touch the circle, is determined by the slope of the tangent lines and the circle's equation, not the radius.
4. Can you use any two tangent lines on a circle to find the point of intersection?
No, you cannot use any two tangent lines on a circle to find the point of intersection. The tangent lines must be drawn from the same external point to the circle. This means that they must have the same slope and be equidistant from the center of the circle.
5. What is the significance of finding the point of intersection between two tangent lines on a circle?
Finding the point of intersection between two tangent lines on a circle can be useful in solving various geometric problems. It can also help in analyzing the behavior of a circle and its tangents, and in proving theorems related to circles and tangents. In addition, this concept is often used in real-life applications such as in engineering and architecture.
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