Polar graph: stuck on sketching a cardioid

**ssadi** · October 17th 2008, 12:56 PM

Sketch the cardioid with polar equation r=2(1+cos theta)
Prove that for all the chords POQ of the cardioid which pass through the pole O, the length of PQ is constant.
How do I do the green part?
I have gone as far as to deduce that if P (r1, theta1) and Q is (r2, theta2)
Then theta2=theta1-180
r2=2+2cos (theta1-180)
There isn't given a formula in the book to find the distance between two polar coordinates, I found one on net:

Here is the sketch of the graph:

I am stuck here for over two hours. Emergency help...I am drow...(Oops I gulped up some water)

**Laurent** · October 17th 2008, 01:13 PM

Originally Posted by ssadi

There isn't given a formula in the book to find the distance between two polar coordinates, I found one on net:

You don't need any formula from the internet... The distance from $O$ to $P(r,\theta)$ is $r$ . So what you have to do is add the values of $r$ for $\theta_1$ and $\theta_2=180-\theta_1$ (in degrees), and check that it does not depend on $\theta_1$ .

**ssadi** · October 17th 2008, 01:54 PM

Originally Posted by Laurent

You don't need any formula from the internet... The distance from $O$ to $P(r,\theta)$ is $r$ . So what you have to do is add the values of $r$ for $\theta_1$ and $\theta_2=180-\theta_1$ (in degrees), and check that it does not depend on $\theta_1$ .

theta2=theta1-180 according to my calculations. I am giving it a try.

**Laurent** · October 17th 2008, 01:57 PM

Originally Posted by ssadi

theta2=theta1-180 according to my calculations. I am giving it a try.

Sorry I made a mistake, in fact $\theta_2=\theta_1+180$ . Because $\theta_2-\theta_1$ is a flat angle.

**ssadi** · October 17th 2008, 01:59 PM

Originally Posted by Laurent

You don't need any formula from the internet... The distance from $O$ to $P(r,\theta)$ is $r$ . So what you have to do is add the values of $r$ for $\theta_1$ and $\theta_2=180-\theta_1$ (in degrees), and check that it does not depend on $\theta_1$ .

It worked. Thanks.

**ssadi** · October 17th 2008, 02:00 PM

Originally Posted by Laurent

Sorry I made a mistake, in fact $\theta_2=\theta_1+180$ . Because $\theta_2-\theta_1$ is a flat angle.

Does that mean I am wrong here? I have verified it with calculator, my formula worked for polar -pi<theta<pi

**Laurent** · October 17th 2008, 02:07 PM

Originally Posted by ssadi

Does that mean I am wrong here? I have verified it with calculator, my formula worked for polar -pi<theta<pi

No, indeed, you're right as well... I considered the graph defined for $0<\theta_2<2\pi$ , while you considered $-\pi<\theta_2<\pi$ , that's why. Forget my previous post, I was correcting my first one, not yours in fact.

**ssadi** · October 17th 2008, 02:10 PM

Originally Posted by Laurent

No, indeed, you're right as well... I considered the graph defined for $0<\theta_2<2\pi$ , while you considered $-\pi<\theta_2<\pi$ , that's why. Forget my previous post, I was correcting my first one.

I guessed so, that your limits might be different. But my problem is solved anyway.

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