Solving a Polar Equation with Negative Numerator: Is It Possible?

In summary, the given polar equation is r = 5/(-1 + 2 cos θ), with the format r = ed/(1 + e cos θ) where e and d are always positive. It is possible to solve this equation by multiplying r by -1/-1, resulting in r = -5/(1 - 2 cos θ). The negative value of r indicates a direction of 180 degrees in the opposite direction, with the point located at 5/3 on the positive x-axis when theta is 180 degrees. This does not indicate a misprint, as r can be negative in polar coordinates.
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I have the given polar equation that I have to graph.

r = 5/(-1 + 2 cos θ)

The format is r = ed/(1 +e cosθ)

(e and d are always positive)

I can solve multiply r by -1/-1 --> r = -5/(1 - 2 cosθ).

Is this possible? I thought that the numerator (ed) cannot be a negative number. Is this a misprint or is there a way to solve this?
 
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  • #2
r can be negative... it just means that you actually go 180 degrees the other direction. So for example, when theta is 180 degrees, r is -5/3... so the point is at 5/3 on the *positive* x-axis.
 

What is a polar equation with a negative numerator?

A polar equation with a negative numerator is an equation in polar coordinates that contains a negative coefficient for the radial term. This means that the radius of the polar coordinates is multiplied by a negative number in the equation.

Can a polar equation with a negative numerator be solved?

Yes, a polar equation with a negative numerator can be solved using standard algebraic techniques. The resulting equation will still be in polar form, but it can be converted to rectangular form if desired.

What are the steps for solving a polar equation with a negative numerator?

The steps for solving a polar equation with a negative numerator are as follows:
1. Simplify the equation by factoring out any common factors.
2. Convert the equation to rectangular form if desired.
3. Solve the equation using standard algebraic techniques.
4. If the equation was converted to rectangular form, convert the solution back to polar form.

Are there any special considerations when solving a polar equation with a negative numerator?

Yes, when solving a polar equation with a negative numerator, it is important to pay attention to the signs of the coefficients and variables. Also, be sure to convert the solution back to polar form if the equation was converted to rectangular form.

What are some real-world applications of solving polar equations with negative numerators?

Polar equations with negative numerators are commonly used in physics and engineering to model circular motion, such as the motion of planets and satellites. They can also be used in navigation and mapping, as well as in computer graphics for creating polar plots.

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