How can I find the center and radius of a circle from two different equations?

[aruwin]

Hi,
how do I find the center and radius from these equations? The 2 equations represent 2 different circles, by the way.

 

[chisigma]

Hi,
how do I find the center and radius from these equations? The 2 equations represent 2 different circles, by the way.

The only visible function and the exponential function...

$\displaystyle e^{z} = \sum_{n=0}^{\infty} \frac{z^{n}}{n!}\ (1)$

... which has centre in z=0 and converges for any value of z...

Kind regards

$\chi$ $\sigma$

 

[aruwin]

The only visible function and the exponential function...

$\displaystyle e^{z} = \sum_{n=0}^{\infty} \frac{z^{n}}{n!}\ (1)$

... which has centre in z=0 and converges for any value of z...

Kind regards

$\chi$ $\sigma$

I forgot to mention that P+Q is a complex number in the form Z= X+iY because P is the real power while Q is the reactive power.

 

[DavidCampen]

I will work the 1st equation. I will eliminate subscripts to make the notation simpler and also let \(\alpha = \theta - \frac{\pi}{2} \) and \(K = \frac{E^2}{X}\)

So the 1st equation becomes:
\(P + Q = K(\,(0.9)^2j + 0.9e^{j\alpha}\,)\)

we know that \(e^{j\alpha} = \cos(\alpha) + j \sin(\alpha)\)

so the 1st equation becomes:
\(P + Q = K[\,(0.9)^2j + 0.9(\,\cos(\alpha) + j \sin(\alpha)\,)]\)

or
\(P + Q = 0.9K(\,\cos(\alpha) + j \sin(\alpha)\,) + (0.9)^2Kj \)

so \(P + Q\) as a function of \(\alpha\) is a circle of radius \(0.9K\) and center (\(0.9)^2Kj \)

 

[blue_raver22]

To find the center and radius of a circle, you can use the general equation for a circle (x-h)^2 + (y-k)^2 = r^2, where (h,k) represents the coordinates of the center and r represents the radius. This equation can be rewritten in the form (x-a)^2 + (y-b)^2 = c, where (a,b) is the center and c is the squared radius.

To find the center, you can compare the equations to this general form and solve for (a,b). Once you have the values for (a,b), you can use the distance formula (d = √(x2-x1)^2 + (y2-y1)^2) to find the radius by plugging in the coordinates of the center and any point on the circle. Alternatively, you can also find the radius by taking the square root of c in the rewritten form of the equation.

Since you have two different equations representing two different circles, you will need to repeat this process for each equation to find the center and radius of each circle. Once you have the values for (a,b) and c for each equation, you can then plot the centers on a coordinate plane and draw the circles with the corresponding radii to visually represent the circles.