# Thread: parametric form of equation

1. ## parametric form of equation

how do i express $x^2+y^2-28x+4y+120=0$ in parametric form

2. Its also $(x-14)^2+(y+2)^2-80 = 0$...

3. Hello, sigma1!

How do i express $x^2+y^2-28x+4y+120\:=\:0$ in parametric form?

We have: . $x^2 - 28x + y^2 + 4y \;=\;-120$

Complete the square: . $x^2 - 28x {\color{blue}\:+\: 196} + y^2 + 4y {\color{red}\:+\: 4} \;=\;-120 {\color{blue}\:+\: 196} {\color{red}\:+\: 4}$

And we have: . $(x-14)^2 + (y+2)^2 \;=\;80$

This is a circle with center $(14,-2)$ and radius $4\sqrt{5}$

The parametric equations are: . $\begin{Bmatrix}x &=& 14 + 4\sqrt{5}\,\cos\theta \\ \\[-3mm] y &=& \text{-}2 + 4\sqrt{5}\,\sin\theta \end{Bmatrix}$

4. i inderstand uto the point where you complete the square but can you please explain the principle you used to arrive at your answer. thanks.

5. In general,

$x^2+y^2 +ax+by +c=0$

$x^2+ 2\frac a2x + (\frac a2)^2+y^2+2\frac b2y +(\frac b2)^2+c-(\frac a2)^2-(\frac b2)^2=0$

$(x-\frac a2)^2+(y-\frac b2)^2=(\frac a2)^2+(\frac b2)^2-c$

which you can check by expanding the squares. In your case, a=-28, b=4, c=120.

http://www.algebralab.org/lessons/le...gthesquare.xml

6. i do understand that point but i dont understand how you use that to get to the parametric equations. with the trig variables.

7. Oh. A circle of radius r centred at the origin is given by the parametric equations

$\begin{Bmatrix}x &=& r\,\cos\theta \\ \\[-3mm] y &=& r\,\sin\theta \end{Bmatrix}$.

This is pretty much just a definition. Can you translate these so that the centre is at the correct point?

8. Originally Posted by sigma1
i do understand that point but i dont understand how you use that to get to the parametric equations. with the trig variables.
a circle centered at the origin, $x^2 + y^2 = r^2$ , can be written parametrically as

$x = r\cos{t}$

$y = r\sin{t}$

shifting the center from the origin to the point $(h,k)$ yields the equations

$x = h + r\cos{t}$

$y = k + r\sin{t}$

... that's all.

9. Originally Posted by skeeter
a circle centered at the origin, $x^2 + y^2 = r^2$ , can be written parametrically as

$x = r\cos{t}$

$y = r\sin{t}$

shifting the center from the origin to the point $(h,k)$ yields the equations

$x = h + r\cos{t}$

$y = k + r\sin{t}$

... that's all.

thanks alot i inderstand it clearly now.