Question

Find the equation of the circle which circumscribes the triangle formed by the lines x + y + 3 = 0, x − y + 1 = 0 and x = 3

Answer 1

In \[∆\]

(i) Let AB represent the line x + y + 3 = 0.       ...(1)
Let BC represent the line x − y + 1 = 0.           ...(2)
Let CA represent the line x = 3.                       ...(3)
Intersection point of (1) and (3) is  \[\left( 3, - 6 \right)\]
Intersection point of (1) and (2) is (−2, −1).
Intersection point of (2) and (3) is (3, 4).
Therefore, the coordinates of A, B and C are  \[\left( 3, - 6 \right)\], (−2, −1) and (3, 4), respectively.

Let the equation of the circumcircle be  \[x^2 + y^2 + 2gx + 2fy + c = 0\]

It passes through A, B and C.

∴ \[45 + 6g - 12f + c = 0\]

\[5 - 4g - 2f + c = 0\]

\[25 + 6g + 8f + c = 0\]

\[\therefore g = - 3, f = 1, c = - 15\]

Hence, the required equation of the circumcircle is 

\[x^2 + y^2 - 6x + 2y - 15 = 0\]