Tutorial on Equation of Circle(3)

This is a continuation of tutorial on equations of circles (1) and (2)



Example 6

Find the equation of the circle that is tangent to the line whose equation is given by x + y = 2 and has its center at (3 , 5).

Solution to Example 6

The first step is to determine the point of tangency of the circle and the line x + y = 2. Use the property of the circles that a line through the center C of a circle and the point of tangency T (let us call this line CT) and the line x + y = 2 (let us call this line LT) tangent to the circle are perpendicular (see graph below).

 line tangent to circle.

Figure 1. line tangent to a circle and perpendicular to line through center.
The slopes m1 and m2 of two perpendicular lines are related by the formula: m1 m2 = - 1. Find the slope m1 of line LT.
x + y = 2
y = - x + 2
m1 = -1
We now use the formula: m1 m2 = - 1 to find the slope m2 of line CT.
m2 = -1 / m1 = 1
The equation of the line CT which passes by the center C(3 , 5) is given by
y - 5 = m2 (x - 3)
y = x + 2
The point of tangency is the intersection of lines CT and LT and is found by solving the system of equations of the two lines.
x + y = 2
y = x + 2
The point of tangency is at (0 , 2).
The distance between the center of the circle and the point of tangency is equal to the radius r of the circle and is given by.
r = √[ (3 - 0)
2 + (5 - 2)2 ] = 3√(2)
Let h and k be the x and y coordinates of the center of the circle and r it radius, the equation of the circle in standard form is given by:
(x - h)
2 + (y - k)2 = r2
(x - 3)
2 + (y - 5)2 = (3 √(2))r2
(x - 3)
2 + (y - 5)2 = 18

Shown below is the graph of the circle and the line x + y = 2 tangent to it.

graph of circle and line tangent to it.

Figure 2. Graph of solution: line tangent to a circle.

Matched Exercise

Find the equation of the circle that is tangent to the line whose equation is given by x + 2y = 2 and has its center at (0,5).

More References and links


Tutorials on equation of circle.
Tutorials on equation of circle (2).
Match Equations of Circles to Graphs. Excellent interactive activity where equations of circles are matched to graphs.
Interactive tutorial on equation of circle.