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 where A and C are nonzero integers.
- Considering the general form , if , 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 is given by where .
The equation of the circle whose center is at is given by where
The equation of the circle whose radius is equal to 1 and center at is given by . This is called the Unit Circle.
Example 1. Find the center and radius of the circle represented by .
Using completing the square, we can solve for the coordinates of the center and the radius.
Hence the center is at with a radius of
Example 2. What is the equation of the circle whose center is at and passes through ?
In this case, we need to solve for the radius using the distance formula:
Thus, the equation of the circle is .
Example 3. What is the equation of the circle that passes through the points ?
Use the general equation of the circle where A = C = 1.
This forms a linear system
Solving the system yields
Substitute the obtained values to the general equation .
Thus, the general equation of the circle is .
The standard form equation is .
More Problems involving Circles, Tangents, and Radical Axis
Example 1. What is the equation of the circle whose center is at a tangent to the line ?
We need to apply the formula of the distance of a point to a line:. This is aimed to determine the distance between the line and the center.
Hence, the equation of the circle is .
Example 2. Find the radical axis of the given circles .
Subtract the two equations. The resulting linear equation is the radical axis.
The radical axis is .
The graph of the two circles and their radical axis is shown below.
- 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 has a center at .
- The circle has a center at .
- In finding the radius of the circle, always take the positive root. For instance, the circle , then .
- The alternative formula to find the radius of the circle is .
- Use completing the square when transforming the general equation of a circle to its standard form .
- In a circle, in the equation .
- In an ellipse, in the equation .
- 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 provided that .
- Do not be confused with the discriminant of an ellipse since it is also . The condition in this case is .
- In solving for the equation of the circle when the three points are given, we use to generate systems of linear equations with D, E, and F as variables.
- Keith Madrilejos
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