SOLUTION: Find the equations of the lines passing through the origin that are tangent to a circle with radius 2 and center at point (2, 1).

Question 1146382: Find the equations of the lines passing through the origin that are tangent to a circle with radius 2 and center at point (2, 1).
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Clearly with a center (2,1) and radius 2, one of the two lines tangent to the circle passing through the origin is x=0.

Some work is needed to find the other line....

Let (a,b) be the other point of tangency to the given circle of a line that passes through the origin. Then

(1) The distance from (2,1) to (a,b) is 2:

(2) The slope of the radius to the point of tangency is
The slope of the tangent line is the negative reciprocal,
The tangent passes through the points (0,0) and (a,b); so an equation of the tangent is

Subtracting the equation in (1) from the equation in (2),

Substituting b=1-2a in (2)...

And then, to find b

The point (a,b) is (4/5,-3/5).

Since the tangent line passes through the origin, the equation of the line is

A graph showing part of the lower half of the given circle and the second tangent line; the first tangent line is of course x=0, the y-axis:

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