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?
Let the circle have equation x^2+y^2=r^2, for given values of A,B,C the system formed by this equation and the line has 0, 1, or 2 solutions depending on the radius of the circle. The circle is tangent to the line if there is exactly one solution.
The system involves a quadratic and so will have one solution if its discriminant is zero.
Substituting the line into the circle gives
x^2 (-Ax/B + C/B)^2 = r^2
(1 + A^2/B^2)x^2 + (-2AC/B^2)x + (C^2/B^2 - r^2)=0
which has discriminant
(-2AC/B^2)^2 - 4(1 + A^2/B^2)(C^2/B^2 - r^2)
Setting this equal to zero and solving for r:
r^2 = C^2/(A^2+B^2)
as given in the previous solutions.
The point of intersection can be found by substituting this in up above.
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Posted by Jer
on 2017-01-05 14:31:47 |
Alternate solution
