Parametric equations for a circle

In summary, parametric equations for a circle are equations that describe the coordinates of points on a circle using one or more parameters. They can be derived using sine and cosine functions, and are useful for describing the motion of a point on a circle and solving problems involving circles. These equations can also be used for other shapes such as ellipses, parabolas, and hyperbolas. To graph them, you can plot the x and y coordinates or use a graphing calculator or software.
  • #1
ILoveBaseball
30
0
The circle [tex] (x-3)^2 + (y-4)^2 = 9[/tex] can be drawn with parametric equations.
Assume the circle is traced clockwise as the parameter increases.

if x = 3+3cos(t) then y= _______?

wouldnt y just be 3+4sin(t)?
 
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  • #2
Notice the circle is drawn clockwise, your parameter is -t.

Also for your y component, you multiply sin(t) by the radius of the circle. The radius of this circle is 3, not 4. so y = 4+3sin(-t) = 4-3sin(t)
 
  • #3


Yes, you are correct. The parametric equations for the circle (x-3)^2 + (y-4)^2 = 9, traced clockwise as the parameter increases, would be:

x = 3+3cos(t)
y = 4+3sin(t)

This is because the center of the circle is at (3,4) and the radius is 3. Therefore, as the parameter t increases, the point (x,y) moves around the circle in a clockwise direction, with x being the horizontal coordinate and y being the vertical coordinate. The equations x = 3+3cos(t) and y = 4+3sin(t) represent the x and y coordinates of the circle at any given point on the circle, as determined by the parameter t.
 

1. What are parametric equations for a circle?

Parametric equations for a circle are a set of equations that describe the coordinates of points on a circle in terms of one or more parameters. These equations are commonly used in mathematics and physics to represent the motion of a point around a circle.

2. How do you derive parametric equations for a circle?

The parametric equations for a circle can be derived by using the sine and cosine functions to represent the x and y coordinates of a point on the circle. The general form of the equations is x = r*cos(t) and y = r*sin(t), where r is the radius of the circle and t is the parameter that represents the angle of rotation.

3. What is the significance of parametric equations for a circle?

Parametric equations for a circle allow us to describe the motion of a point on a circle in a simple and concise way. They are also useful in solving problems involving circles in mathematics and physics, such as finding the area or circumference of a circle.

4. Can parametric equations for a circle be used to represent other shapes?

While parametric equations are commonly used for circles, they can also be used to represent other shapes such as ellipses, parabolas, and hyperbolas. The specific equations used will vary depending on the shape being represented.

5. How do you graph parametric equations for a circle?

To graph parametric equations for a circle, you can plot the x and y coordinates for various values of the parameter t. This will result in a circular shape, with the radius and center determined by the equations used. Alternatively, you can also use a graphing calculator or software to plot the equations and generate a graph.

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