perplexus.info :: Geometry : From Tangent to Circle

From Tangent to Circle (Posted on 2017-01-04)

The line Ax+By=C (with C nonzero) is tangent to some circle centered at the origin.

What is the radius of that circle in terms of A, B, and C?
What is the point of tangency in terms of A, B, and C?

No Solution Yet Submitted by Brian Smith

Rating: 3.0000 (1 votes)

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solution for part 2; outline for part 1

| Comment 1 of 4

ax + by = c

by= c-ax

y= (c-ax)/b

slope = -a/b

perpendicular line through origin:

y = bx/a

which intersects original line by solving

ax + (b^2)x/a = c

(a+(b^2)/a)x = c

x = c/(a+(b^2)/a) = ca / (a^2+b^2)

y = bca / (a*(a^2+b^2))

Point of tangency:

(ca / (a^2+b^2), bca / (a*(a^2+b^2))

The radius of the circle can be found from the pythagorean theorem applied to these x and y coordinates.

Edited on January 5, 2017, 12:09 pm

Posted by Charlie on 2017-01-04 13:38:09

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