Equation of a circle question?

Question

how do I derive the equation of a circle, given this information. The circle passes thorugh the origin, and intercepts the x axis at 1 and the y axis at 2.

Answer 1

So 3 points on it are (0,0), (1,0) and (0,2)

Its equation is (x-h)^2 + (y-k)^2 = r^2

Plugging in each point for x and y in this equation gives three equations with the three variables h, k and r so you can solve for them:

(0-h)^2 + (0-k)^2 = r^2

(1-h)^2 + (0-k)^2 = r^2

(0-h)^2 + (2-k)^2 = r^2

h^2 + k^2 = r^2

1 - 2h + h^2 + k^2 = r^2

are the first two: multiply the first by -1 and add it to the second

getting 1 - 2h = 0 so h = 1/2

The third simplifies to h^2 + 4 - 4k + k^2 = r^2; do the same thing here

getting 4 - 4k = 0 so k = 1

Then plug those in to the first equation to get r

(1/2)^2 + 1^2 = r^2

1/4 + 1 = 5/4 = r^2

so the equation is (x - 1/2)^2 + (y - 1)^2 = 5/4

Or you could graph the 3 points, see that they are 3 vertices of a rectangle, plot the fourth vertex (1,2) then draw the diagonals to find the center and radius