Precalculus - Circles

Brief Introduction to Conic Sections and Circles

  • The figures are a group of curves called Conic Sections. They are formed from the intersection of a plane and right circular cones. Notice in the figures that they do not intersect the vertex of the cone. 
  • Basically, there are four types of conic sections - Circle, Parabola, Ellipse, and Hyperbola.
  • In general, conic sections are represented by second-degree equations. This is written as Ax2+Bxy+Cy2+Dx+Ey+F=0 where A and C are nonzero integers.
  • Considering the general form Ax2+Bxy+Cy2+Dx+Ey+F=0, if B=0 and A=C, then the graph of this equation represents a CIRCLE.
  • A circle is a locus moving in a plane in such a way that its distance from a fixed point called the center remains constant. 
  • A circle is a special kind of ellipse.
  • A circle is formed when a cone is cut perpendicular to its axis.

The equation of the circle where the center is  0,0 is given by x2+y2=r2 where r=radius

The equation of the circle whose center is at Ch,k is given by x-h2+y-k2=r2 where r=radius

The equation of the circle whose radius is equal to 1 and center at 0,0 is given by x2+y2=1. This is called the Unit Circle.

Solved Examples

Example 1. Find the center and radius of the circle represented by x2-8x-6y+y2-11=0.

Solution:

Using completing the square, we can solve for the coordinates of the center Ch,k and the radius.

x2-8x+y2-6y-11=0

x2-8x+16+y2-6y+9=11+16+9

x-42+y-32=36

Hence the center is at C4,3 with a radius of r=10

 

Example 2. What is the equation of the circle whose center is at C5,7 and passes through 2,8?

Solution:

 In this case, we need to solve for the radius using the distance formula:

r2=5-22+7-82

r2=9+1r=10

Thus, the equation of the circle is x-52+y-72=10.

 

Example 3. What is the equation of the circle that passes through the points -3,-9, 2,-4, and 6,-6?

Solution:

Use the general equation of the circle Ax2+Cy2+Dx+Ey+F=0 where A = C = 1.

This forms a linear system -3D-9E+F=-902D-4E+F=-206D-6E+F=-72

Solving the system yields D=-4, E=18, and F=60

Substitute the obtained values to the general equation x2+y2+Dx+Ey+F=0.

Thus, the general equation of the circle is x2-4x+y2+18y+60=0.

The standard form equation is x-22+y+92=25.

More Problems involving Circles, Tangents, and Radical Axis

Example 1. What is the equation of the circle whose center is at C-3,-4 a tangent to the line 3x+4y=15?

Solution:

We need to apply the formula of the distance of a point to a line:d=Ax1+By1+CA2+B2. This is aimed to determine the distance between the line and the center.

d=3-3+4-4-1525d=8

Hence, the equation of the circle is x+32+y+42=64.

 

Example 2. Find the radical axis of the given circles x2+y2=4 and x-22+y+92=25.

Solution:

Subtract the two equations. The resulting linear equation is the radical axis.

x2-4x+y2+18y=-60-x2+y2=4

The radical axis is 2x-9y=32.

The graph of the two circles and their radical axis is shown below.

Cheat Sheet

  • Be careful in dealing with the signs when the goal is to find the coordinates of the center when the equation of the circle is given.
  • For instance, the circle x+82+y-32=15 has a center at C-8,3
  • The circle x-72+y-12=49 has a center at C7,1.
  • In finding the radius of the circle, always take the positive root. For instance, the circle x+122+y-142=16, then r2=16r=4.
  • The alternative formula to find the radius of the circle is r2=B2+C2-4AD4A2.
  • Use completing the square when transforming the general equation of a circle to its standard form x-h2+y-k2=r2

Blunder Areas

  • In a circle, A=C in the equation Ax2+Cy2+Dx+Ey+F=0.
  • In an ellipse, AC>0 in the equation Ax2+Cy2+Dx+Ey+F=0
  • When the equation of a conic section is given, always know the difference between a circle and an ellipse.
  • The discriminant of a circle is given by B2-4AC<0 provided that A=C and B=0
  • Do not be confused with the discriminant of an ellipse since it is also B2-AC<0. The condition in this case is AC>0.
  • In solving for the equation of the circle when the three points are given, we use Ax2+Cy2+Dx+Ey+F=0 where A=C=1 to generate systems of linear equations with D, E, and F as variables.