Question

Find the equation of the circle which circumscribes the triangle formed by the lines
(i) x + y + 3 = 0, x − y + 1 = 0 and x = 3
(ii) 2x + y − 3 = 0, x + y − 1 = 0 and 3x + 2y − 5 = 0
(iii) x + y = 2, 3x − 4y = 6 and x − y = 0.

Answer 1

In $∆$ABC:
(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 $x^2+y^2-6x+2y-15=0$.


(ii) In $∆$ABC:
Let AB represent the line 2x + y − 3 = 0. ...(1)
Let BC represent the line x + y − 1 = 0. ...(2)
Let CA represent the line 3x + 2y − 5 = 0. ...(3)

Intersection point of (1) and (3) is (1, 1).
Intersection point of (1) and (2) is (2, −1).
Intersection point of (2) and (3) is (3, −2).

The coordinates of A, B and C are (1, 1), (2, −1) and (3, −2), respectively.

Let the equation of the circumcircle be $x^2+y^2+2gx+2fy+c=0$.
It passes through A, B and C.

∴ $2+2g+2f+c=0$
$5+4g-2f+c=0$
$13+6g-4f+c=0$

$\therefore g=\frac{-13}{2},f=\frac{-5}{2},c=16$

Hence, the required equation of the circumcircle is $x^2+y^2-13x-5y+16=0$.

(iii) In $∆$ABC:
Let AB represent the line x + y = 2. ...(1)
Let BC represent the line 3x − 4y = 6. ...(2)
Let CA represent the line x − y = 0. ...(3)

Intersection point of (1) and (3) is (1, 1).
Intersection point of (1) and (2) is (2, 0).
Intersection point of (2) and (3) is (−6, −6).

The coordinates of A, B and C are (1, 1), (2, 0) and (−6, −6), respectively.

Let the equation of the circumcircle be $x^2+y^2+2gx+2fy+c=0$.
It passes through A, B and C.
∴ $2+2g+2f+c=0$
$4+4g+c=0$
$72-12g-12f+c=0$

$\therefore g=2,f=3,c=-12$

Hence, the required equation of the circumcircle is $x^2+y^2+4x+6y-12=0$.