# Parametric equations to rectangular equations.

• Apr 5th 2009, 09:23 PM
cheet0face
Parametric equations to rectangular equations.
I know how to change a rectangular equation into a parametric one but can't do it vice versa.

An example is: (x+2)^2/25-(y-5)^2/12=1

I'm flabbergasted!
• Apr 5th 2009, 10:27 PM
Hello cheet0face
Quote:

Originally Posted by cheet0face
I know how to change a rectangular equation into a parametric one but can't do it vice versa.

An example is: (x+2)^2/25-(y-5)^2/12=1

I'm flabbergasted!

There are no hard and fast rules about doing it this way round, but the equation you mention is a fairly standard one - it's a hyperbola, and a variation on the simpler version:

$\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$

except that its centre has been moved from the origin to $(-2, 5)$.

We use trig functions to write equations like this in parametric form. In this case we can use the fact that $\sec^2\theta - \tan^2\theta= 1$, by writing:

$x=5\sec \theta - 2$ and $y = 2\sqrt3\tan\theta + 5$

You'll see that if you eliminate $\theta$ between these two equations you get:

$\frac{x+2}{5} = \sec\theta$ and $\frac{y-5}{2\sqrt3}=\tan\theta$

So, using the above trig identity:

$\frac{(x+2)^2}{25} - \frac{(y-5)^2}{12} = 1$