Well I'm kinda stumped and I need you guys...
1.) The length of a diameter of a circle described by 9x^2 + 9y^2 = 16 is?
2.) Find the equation of the circle with center on x+y = 4 and 5x+2y+1=0 and having a radius of 3.
1)
$\displaystyle 9x^2 + 9y^2 = 16$
$\displaystyle x^2 + y^2 = \frac{16}{9}$
As general form is $\displaystyle x^2 + y^2 = r^2$
$\displaystyle \implies r^2 = \frac{16}{9}$
$\displaystyle r = \frac{4}{3}$
Diameter is two times radius.
$\displaystyle \therefore d=\frac{8}{3}$
Regarding #2:
The equation for a circle with center (h,k) and radius r is:
$\displaystyle (x-h)^2 + (y-k)^2 = r^2$.
So you know r = 3. Where's the center?
Recall that it lies on the two lines given, so find the intersection of the two lines. I did it by substitution, starting with the first equation, which gives:
$\displaystyle y=4-x$.
Sub this into the other equation for y, which gives:
$\displaystyle 5x+8-2x+1=0$
$\displaystyle 3x+9=0 $
$\displaystyle 3x=-9$
$\displaystyle x=-3$.
Recall $\displaystyle y=4-x$, so $\displaystyle y=4-(-3)=7$.
So your center is at $\displaystyle (-3,7)$. Now you know h, k, and r, the three essentials for writing the equation of a circle.
There you are!