How to Find the Equation of a Normal Line to a Circle

Lines normal to a circle intersect the circle's center.

A line that is normal to a circle is perpendicular to the circle where the two intersect. Such a line goes through the center of the circle. Therefore, you can develop an equation describing a normal line by using both the point of intersection between the circle and normal line and the coordinates of the center of the circle.

Instructions

1. Identify the coordinates of the center of the circle. Denote these coordinates as (x0,y0).

   Consider, for example, that the center of the circle (x-2)^2 + (y+3)^2 = 25 is (2,-3).

2. Identify the coordinates of the point of intersection between the normal line and the circle. Denote these coordinates as (x1,y1). The coordinates of intersection will likely be given in the problem you're solving.

3. Determine the slope, m, of the line by solving m = (y1-y0)/(x1-x0).

   Continuing with the example from Step 1, suppose the point of intersection is (x1,y1) = (2+5/√2,-3+5/√2). Then m = (2+5/√2 - 2)/(-3+5/√2 - (-3)) = (5/√2) / (5/√2) = 1. So the slope of the normal line is 1.

4. Complete the slope-intercept formula, y=mx+b, by entering the center of the circle in the formula to calculate the y-intercept.

   Continuing with the example, y=mx+b becomes -3=1*2+b. So b=-5. So the general formula for the normal line y=mx+b becomes y=x-5.

References

• Villanova University: Normal Lines
• Analyze Math: Equation of a Circle