change the polar equation r=5/1+cos(theta) to rectangular form.
How do I do this? Thank you! :)
x^2 + y^2 = (5 - x) square right side since x^2 + y^2 = r^2
x^2 + y^2 = 25 - 10x + x^2 the x^2 cancel out
-x^2 -x^2'
y^2 = 25 - 10x is the answer
Well, I'm glad you asked! Converting polar equations to rectangular form can be quite entertaining. To convert \(r = \frac{5}{1 + \cos(\theta)}\) to rectangular form, we can use some nifty mathematical sleight of hand.
First, recall that \(r\) represents the distance from the origin to a point in polar coordinates. To convert to rectangular coordinates, we'll need to express \(r\) in terms of \(x\) and \(y\).
We know that \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). So we're going to replace \(r\) in the equation with its equivalent expressions.
Let's start doing some tricky math! We'll multiply both sides of the equation by \(1 + \cos(\theta)\) to get rid of the denominator:
\(r(1 + \cos(\theta)) = 5\).
Expanding this gives us \(r + r\cos(\theta) = 5\).
Now, substituting \(x\) for \(r\cos(\theta)\), we can simplify further:
\(x + y = 5\).
And there you have it! The rectangular form of the polar equation \(r = \frac{5}{1 + \cos(\theta)}\) is simply \(x + y = 5\).
I hope that brought a smile to your face! If you have any more questions, feel free to ask!
To change the polar equation r = 5/(1 + cos(theta)) to rectangular form, we can use the following relationships between polar and rectangular coordinates:
x = r * cos(theta)
y = r * sin(theta)
Let's substitute these expressions into the given polar equation:
x = (5/(1 + cos(theta))) * cos(theta)
y = (5/(1 + cos(theta))) * sin(theta)
Simplifying these equations, we get:
x = 5 * cos(theta) / (1 + cos(theta))
y = 5 * sin(theta) / (1 + cos(theta))
Therefore, the rectangular form of the polar equation r = 5/(1 + cos(theta)) is:
x = 5 * cos(theta) / (1 + cos(theta))
y = 5 * sin(theta) / (1 + cos(theta))
Keep in mind that the rectangular form may not always simplify further, and it is perfectly acceptable to leave it in this form.
recall:
sinØ = y/r
cosØ = x/r
r^2 = x^2 + y^2
I will assume you mean:
r=5/(1+cos(theta) )
r=5/(1+x/r)
r + x = 5
√(x^2 + y^2) = 5-x
confirmation:
http://www.wolframalpha.com/input/?i=%E2%88%9A%28x%5E2+%2B+y%5E2%29+%3D+5-x
http://www.wolframalpha.com/input/?i=+polar+plot+r%3D5%2F%281%2Bcos%28theta%29+%29+from+-4%CF%80+to+4%CF%80
r=5/1+cos theta
There's a nice discussion of this topic at
http://laurashears.info/math122/unit4/polarAndParamFormsOfParabola/