Show proof of point C in the given problem that involves Polar equation

In summary, to show proof of point C in the given problem involving a polar equation, one must demonstrate that the coordinates derived from the polar equation satisfy the conditions of point C. This involves substituting the polar coordinates into the equation, simplifying, and verifying that the resulting values are consistent with the characteristics of point C as defined in the problem.
  • #1
chwala
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Homework Statement
See attached. My interest is on part C (Highlighted in yellow)
Relevant Equations
polar equations
1712784862585.png


c
1712784886609.png



Parts (a) and (b) are okay ... though the challenge was on part (a)

My graph had a plot of r on the y-axis vs θ on the x-axis). The sketch of my graph looks like is shown below;
1712793020305.png

I suspect the ms had θ on the x-axis vs r on the y-axis.

I used the equation ##r=\sqrt{\dfrac {1}{θ^2+1}}## with various values of ##θ## in the given domain and ended up with a graph opening on the right side of the first and fourth quadrant which is different from the attached graph from the mark scheme.

The shapes were similar.

1712785051115.png



Now for part (c), the steps are quite straightforward, I just want to check why they used ##r\sin θ##. Most probably its the distance of the maximum point of the graph from the x-axis.

The other working steps are quite clear- they used the quotient rule and then sign change to wrap up the question.

I am aware that cartesian to polar form we have ##x = r \cos θ## and ##y = r \sin θ## and therefore to determine distance in the ##y## direction one has to use ##y = r \sin θ##. If this answers my own query then thanks in advance.



1712785469242.png
 
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  • #2
I don't follow your graph of ##r^2 = \frac 1 {\theta^2 + 1}##. If ##\theta = 0##, then ##r = \pm 1##, so you would get points on the horizontal axis 1 unit to the right and 1 unit to the left. Also, if ##\theta = \pi/2##, the values of r would be ##\pm\frac 1 {\sqrt{(\pi/2)^2 + 1}}##, and not ##\pm 1## as you show on your graph.
chwala said:
I suspect the ms had θ on the x-axis vs r on the y-axis.
That's not the way that polar graphs work. Angles are measured in the CCW direction from the positive x axis (the ray ##\theta = 0##), and r is measured out along the ray defined by the angle.
 
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  • #3
Mark44 said:
I don't follow your graph of ##r^2 = \frac 1 {\theta^2 + 1}##. If ##\theta = 0##, then ##r = \pm 1##, so you would get points on the horizontal axis 1 unit to the right and 1 unit to the left. Also, if ##\theta = \pi/2##, the values of r would be ##\pm\frac 1 {\sqrt{(\pi/2)^2 + 1}}##, and not ##\pm 1## as you show on your graph.
That's not the way that polar graphs work. Angles are measured in the CCW direction from the positive x axis (the ray ##\theta = 0##), and r is measured out along the ray defined by the angle.
Noted on the direction of r and the angle. Cheers.
 
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  • #4
Mark44 said:
I don't follow your graph of ##r^2 = \frac 1 {\theta^2 + 1}##. If ##\theta = 0##, then ##r = \pm 1##, so you would get points on the horizontal axis 1 unit to the right and 1 unit to the left. Also, if ##\theta = \pi/2##, the values of r would be ##\pm\frac 1 {\sqrt{(\pi/2)^2 + 1}}##, and not ##\pm 1## as you show on your graph.
That's not the way that polar graphs work. Angles are measured in the CCW direction from the positive x axis (the ray ##\theta = 0##), and r is measured out along the ray defined by the angle.
Mark, forgot your Latex after 16 years in PF???
 
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FAQ: Show proof of point C in the given problem that involves Polar equation

What is a polar equation?

A polar equation is a mathematical representation of a curve in the polar coordinate system, which uses a distance from a reference point (the pole) and an angle from a reference direction (usually the positive x-axis) to define points in a plane. In polar coordinates, a point is represented as (r, θ), where r is the radius and θ is the angle.

How do I convert a polar equation to Cartesian coordinates?

To convert a polar equation to Cartesian coordinates, you can use the relationships x = r*cos(θ) and y = r*sin(θ). By substituting r and θ from the polar equation into these formulas, you can express the equation in terms of x and y.

What does "proof of point C" refer to in a polar equation problem?

How do I determine if a point is on a polar curve?

To determine if a point is on a polar curve, convert the point's Cartesian coordinates (x, y) to polar coordinates (r, θ) using r = √(x² + y²) and θ = arctan(y/x). Then, substitute these values into the polar equation to see if the equation holds true.

What techniques can be used to prove properties of polar equations?

Techniques to prove properties of polar equations include graphical analysis, calculus (such as finding derivatives to analyze slopes and tangents), symmetry considerations, and converting to Cartesian coordinates to leverage algebraic methods. Each technique can provide insights into the behavior and characteristics of the polar curve.

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