A circle has a center that falls on the line y = 2/9x +8 and passes through ( 2 ,5 ) and (1 ,4 ). What is the equation of the circle?

1 Answer
Jan 19, 2018

Equation of the circle is

color(blue)((x - (28/11))^2 + (y - (38/11))^2 = 1.639)

Explanation:

Given
y = (2/9)x + 8 and the line is passing through the center.

Let (h, k) be the coordinates of the circle center.

9h - 2k = 16 Eqn (1)

Standard Equation of a circle is

(x-h)^2 + (y-k)^2 = r^2
Where (h,k) coordinates of the center and r the radius.

enter image source here

(2,5) & (1,4) are points on the circumference of the circle and hence

(2-h)^2 + (5-k)^2 = r^2 = (1-h)^2 + (4-k)^2

4 - 4h + cancel(h^2) + 25 -10k + cancel(k^2) = 1 - 2h + cancel(h^2) + 16 - 8k + cancel(k^2)

4 + 25 - 1 - 16 = -2h + 4h - 8k + 10k

2h + 2k = 12

h + k = 6 Eqn (2)

Solving Eqns (1), (2) we get the coordinates of the center.

O (28/11, 38/11)

r = sqrt((2-(28/11))^2 + (5-(38/11))^2) ~~ 1.639

Equation of the circle is

color(blue)((x - (28/11))^2 + (y - (38/11))^2 = 1.639)