How do you find the standard equation of a circle when all you know is?

Question

How do you find the standard equation of a circle when all you know is...

a. when all you know is the equation of a line passing through the center and a point on the circumference

b. the eqt. of the tagent line, the where the circle and tagent line meet, and a point the circumference passes through

Answer 1

a. the circle is not unique in this case.
b. Write the equation of a line perpendicular to the tangent, and force it to pass through the tangent's point of contact. This make that line one of the radii. Next, assume that O is the center of the circle. Then, for an arbitrary radius R, write the equation of the circle with center O. Next use the fact that O lies on the radius (whose eqn was obtained above) and that the circle should pass through the known point on the cirumference.

Answer 2

Draw a line from the point perpendicular to the line. This line will be the radius of the circle, and the intersection of the perpendicular to the given line will be the center of the circle.

You can find the equation of the perpendicular line by using a slope that is the negative reciprocal of the given line and the coordinates of the given point. Find the intersection of the perpendicular lines which gives you the center. The distance from the center to the given point is the radius.


Using a slope that is the negative reciprocal of the tangent line and the coordinates of the point of tangency you can derive the equation of the line passing throgh the point of tangency and the center of the circle.

Now the problem is exactly the same as problem a.

Answer 3

x + y = number

x,y is the center

take the number and take the square root to get your radius