Circles and Lines
By Ryan Ngo and Colby Kim
Equations and Graph
Behavior
When changing the "a" variable in the equations the diameter of the circle is effected.
What to look for:
- As the "a" increases from 1 to 3, the diameter increases from 1 to 3 simultaneously
When changing the "c" variable in the equations, the circle is rotated around the origin
What to look for:
- The C variable determines the amount of degrees in radians that the circle rotates around the origin.
Restrictions?
- There are no restrictions on the a value of both equations, so the behavior of the graph and r is not different from what it is originally meant to be.
Converting from Polar Form to Cartesian Form
First Equation:
Work for conversion:
Second Equation:
Work for Conversion:
What if We plugged in Values.......?
...For the Circle?
As described earlier, the "a" value determines the diameter of the circle. As can be seen here, the value of a is -3. Therefore, the outermost point of the circle is 3 units away from the other outermost point exactly across from it. The negative flips the circle across the y axis.
On the other hand, the "c" value determines the degree of rotation that the circle has in relation to the unit circle. The degrees are measured in radians. Here, value c is π radians. As can be seen , the circle is oriented in that direction.
...For the line?
While the "a" variable affects the diameter of the circle, it affects the position of the line on the y axis. Here, a is equal to 2, meaning if the line was laid horizontally flat (c=π), it would be 2 units above the x-axis.
The "c" value has a similar effect on the circle as it does the line, that being the degree of rotation in terms of the unit circle The line is oriented in the (+ π/4) direction when c is equal to π/4.
Relationship Between Our Equations
No matter what degree of rotation we turn the circle (c), how long the diameter is, or their location on the polar plane r = a sec (θ — c) will always be the tangent of r = cos (θ-c) as long as the a and c in both equations are equal.
Why are the graphs shaped the way they are?
Since the properties of a circle state that it will always produce repeating r values, changing the values of a, c, and θ will never have an effect on the r value. As a result, the circle ends up going back to the starting point when θ = 2π.
Because this secant is a linear function, it will retain its straight shape regardless of where it is on the polar plane and what values of a, c, and θ are plugged into the equation. The only thing changed is the direction the line is sloped in and its horizontal and vertical shifts.
