Write an equation in standard form for:
a circle with center (0,0) passing through (-3,4).

First, you use the two points to give you a length, and this will serve as our radius. So,
Length = ((4-0)^2+(-3-0)^2)^1/2
Length = (4^2-3^2)^1/2
Length = (16+9)^1/2
Length = 25^1/2
Length = 5 units.
Then, we can use the co-ordinates to figure out the other parts of the circle's equation, as the equation of a circle follows the pattern:
"X^2+Y^2+2GX+2FY+C=0",
the co-ordinates for the centre is (-G,-F) and another way to find the length of the radius is "(G+F-C)^1/2". So, we find out that G and F are 0, and so far we have
"X^2+Y^2+C=0" We can find C by using the second radius formula and our length. So:
(0+0-C)^1/2=5, and hence the square root of -C is equal to five. By squaring both sides we find that C=-25. The circle's final equation is:
"X^2+Y^2-25=0"

It would be the second one because it passes the vertical line test