SOLUTION: write the equation of the circle in general form with a radius r = 5 if the line x- 2y - 1 = 0 is a tangent to this circle at a point (3,1)

Question 769839: write the equation of the circle in general form with a radius r = 5 if the line x- 2y - 1 = 0 is a tangent to this circle at a point (3,1)
Answer by josgarithmetic(39618) (Show Source): You can put this solution on YOUR website!
Start with the given line and the included point, and find the line which is perpendicular at that given point.

the given line.

The line perpendicular to the given line will have slope -2, which is negative reciprocal of the (1/2). We not yet know the y intercept, b.

which must contain the point given, (3,1).

The line we look for is then

We have a set of points based on this wanted and found line, (x, -2x+7)
and a particular point, (3, 1) and we want the distance between them to be 5, which is the radius of the wanted circle. Note again, (3,1) is a point on this circle.

DISTANCE FORMULA!

Square both sides...

____________This will give an x-coordinate for the center of the circle.

Use general solution to quadratic formula to find the value of the x-coordinate.

or

Use the line equation found, , to find either or both y-coordinates for the center of the circle.
x=2:
y=-2*2+7
y=3
so one possible center point is (2,3).
x=4:
y=-2*4+7
y=-1
so other possible center point is (4, -1).

You asked for help including "general form" for the circle, but I'll just finish this in "standard form", and you can convert if you want.

Standard form of your circle is in which the center point is (h, k). You have two possible points to choose from.
Either Or
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