Circumcircle Equation of Triangle with Given Orthocenter and Side Lines

To find the equation of the circumcircle of $\Delta$, we use the property that the reflection of the orthocenter $H(1, 1)$ across any side of a triangle lies on its circumcircle.

1. Intersection of given sides: The intersection of the sides $y = 0$ and $x + y + 1 = 0$ gives one vertex of the triangle: $A = (-1, 0)$

2. Reflections of the orthocenter $H(1, 1)$:

• The reflection of $H(1, 1)$ about the line $y = 0$ is $H_1 = (1, -1)$.
• The reflection of $H(1, 1)$ about the line $x + y + 1 = 0$ is calculated as $H_2 = (-2, -2)$.

1. Equation of the circumcircle: Since the circumcircle passes through $A(-1, 0)$, $H_1(1, -1)$, and $H_2(-2, -2)$, we substitute these points into the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$:

• For $A(-1, 0)$: $1 - 2g + c = 0$
• For $H_1(1, -1)$: $2 + 2g - 2f + c = 0$
• For $H_2(-2, -2)$: $8 - 4g - 4f + c = 0$

Solving this system of linear equations yields $g = \frac{1}{2}$, $f = \frac{3}{2}$, and $c = 0$.

Thus, the required equation of the circumcircle is: $x^2 + y^2 + x + 3y = 0$

Correct Option: (B)