Question 10806: Dear Sir/Madam,
I am having difficulty finding the equation to the following problem.
The center of a circle is (1,2). A line touches the circle at a point (1,4). The line is tangent to a circle at that point. Find the equation of the line.
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! Try this:
If the center of the circle is at (1, 2) and a line is tangent to the circle at point (1, 4), then:
a) The radius of the circle is 2 and,
b) The equation of the line is y = 4.
The point of tangency has the same x-coordinate as the center of the circle, this means that the line has zero slope.
A line that has zero slope and passes through the point (1, 4) has the a y-intercept of (0, 4) and the equation is:
y = (0)x + 4 or y = 4
I had some additional thoughts on this problem. While they won't change my solution, I think they might add a little to the explanation.
Consider this:
The line segment from the center of the circle (1, 2) to the point of tangency (1, 4) is the radius of the circle and has a slope of:
 This means, of course, the this line segment has an undefined slope, i.e., it's vertical.
Now the line is tangent to the circle at the point (1, 4) and, as you know, lines that are tangent to a circle at a given point are perpendicular to the radius at that point. And lines that are perpendicular have slopes that are the negative reciprocal of each other.
The negative reciprocal of 4/0 is -0/4 or just 0.
Therefore, the line that is tangent to the circle in the problem has a slope of zero and it passes through the point (1, 4), its equation is: y = 4
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