# Thread: Determine the equation of a circle

1. ## Determine the equation of a circle

Determine the standard equation for the circle
C that passes through
the points
A
(2, 0) and B(0, 4), and has centre lying on the line L : x+2y+4 = 0.

Hint: First find the centre of C ; it has the same distance from A and B.

Thanks for at least looking at the question

2. Originally Posted by mamt6
Determine the standard equation for the circle
C that passes through
the points
A
(2, 0) and B(0, 4), and has centre lying on the line L : x+2y+4 = 0.

Hint: First find the centre of C ; it has the same distance from A and B.

Thanks for at least looking at the question
let point C be (x,y) ...

$\displaystyle AC = BC$

$\displaystyle (x-2)^2 + (y-0)^2 = (x-0)^2 + (y-4)^2$

$\displaystyle x^2 - 4x + 4 + y^2 = x^2 + y^2 - 8y + 16$

$\displaystyle -4x + 4 = -8y + 16$

$\displaystyle -x + 1 = -2y + 4$

$\displaystyle x = 2y-3$

since (x,y) lies on the line $\displaystyle x+2y+4=0$ ... $\displaystyle x = -2y-4$

so ...

$\displaystyle -2y-4 = 2y-3$

solve for the y-coordinate of point C , then find its x-coordinate.

then determine the equation of the circle ...

$\displaystyle (x - x_c)^2 + (y - y_c)^2 = r^2$