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Getty's question at Yahoo! Answers regarding the distance from a circle where two tangent lines meet

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MarkFL

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Feb 24, 2012
13,775
Here is the question:

How can i find where two tangent lines intersect on a circle?

I need math help tonight. Is there a process i need to follow to find out where the lines meet?
I have posted a link there to this topic so the OP can see my work.
 
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MarkFL

Administrator
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Feb 24, 2012
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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:

getty.jpg

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:

\(\displaystyle r^2+(x-d)^2+r^2-x^2=d^2\)

\(\displaystyle 2r^2+x^2-2xd+d^2-x^2=d^2\)

\(\displaystyle r^2-xd=0\)

\(\displaystyle d=\frac{r^2}{x}\)