Convert the polar equation to the Cartesian equation

In summary, polar coordinates use distance and angle while Cartesian coordinates use x and y coordinates to locate points in a plane. To convert a polar equation to a Cartesian equation, use x = r * cos(theta) and y = r * sin(theta). There are some restrictions when converting between the two coordinate systems, such as polar coordinates not being able to represent vertical lines and Cartesian coordinates not being able to represent circles centered at the origin. To graph a polar equation in the Cartesian coordinate system, convert it to Cartesian form and plot points by substituting values for x and solving for y. Not all polar equations can be converted to Cartesian equations, as some have no equivalent Cartesian form.
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Homework Statement


Replace the polar equation with an equivalent Cartesian equation.

##r^2 = 26r cos θ - 6r sin θ - 9##
a)##(x - 13)^2 + (y + 3)^2 = 9##
b)##(x + 26)^2 + (y - 6)^2 = 9##
c)##26x - 6y = 9##
d)##(x - 13)^2 + (y + 3)^2 = 169##

Homework Equations


##x= r cos \theta##
##y= r sin \theta##
##x^2+y^2 = r^2##

The Attempt at a Solution


##r^2 = 26r cos θ - 6r sin θ - 9##
##x^2 + y^2 = 26 x - 6 y -9 ##
##x^2 -26x + y^2 + 6 y = -9 ##
##(x-13)^2 - 169 + ( y+3 )^2 -9 = -9##
##(x-13)^2 + (y+3)^2 = 169 ##
The answer is (d) . Right ?
 
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  • #2
Looks good.
 
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1. What is the difference between polar and Cartesian coordinates?

Polar coordinates use a distance from the origin and an angle from a reference direction to locate a point in a plane, while Cartesian coordinates use x and y coordinates to locate a point in a plane.

2. How do you convert a polar equation to a Cartesian equation?

To convert a polar equation to a Cartesian equation, use the following formulas: x = r * cos(theta) and y = r * sin(theta), where r is the distance from the origin and theta is the angle from the reference direction.

3. Are there any restrictions or limitations when converting between polar and Cartesian coordinates?

Yes, there are some restrictions when converting between polar and Cartesian coordinates. For example, polar coordinates cannot represent vertical lines, and Cartesian coordinates cannot represent circles centered at the origin.

4. How do you graph a polar equation in the Cartesian coordinate system?

To graph a polar equation in the Cartesian coordinate system, first convert the equation to Cartesian form. Then, plot points by substituting different values for x and solving for y. Finally, connect the points to create the graph.

5. Can all polar equations be converted to Cartesian equations?

No, not all polar equations can be converted to Cartesian equations. Some polar equations, such as r = 1 + cos(theta), have no equivalent Cartesian form.

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