Question

The circumcenter of the triangle formed by lines $xy + 2x + 2y + 4 = 0$ and $x + y + 2 = 0$ is ______.

• A. (0, 0)
• B. (-2, -2)
• C. (-1, -1)
• D. (-1, -2)

Hint:

Right-angled triangle properties are a goldmine for shortcuts!
Circumcenter = Midpoint of the hypotenuse.
Orthocenter = The exact vertex containing the $90^\circ$ right angle.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Factorize the pair of lines to identify the vertices of the triangle.

Step 2: Formula Application:
$xy + 2x + 2y + 4 = (x+2)(y+2) = 0$. The lines are $x = -2$ and $y = -2$.

Step 3: Explanation:
The three lines are $x = -2$, $y = -2$, and $x + y = -2$. - Intersection of $x=-2$ and $y=-2$ is $V_1(-2, -2)$. - Intersection of $x=-2$ and $x+y=-2$ is $V_2(-2, 0)$. - Intersection of $y=-2$ and $x+y=-2$ is $V_3(0, -2)$. This is a right-angled triangle at $(-2, -2)$. In a right-angled triangle, the circumcenter is the midpoint of the hypotenuse. Midpoint of $V_2V_3 = \left( \frac{-2+0}{2}, \frac{0-2}{2} \right) = (-1, -1)$.

Step 4: Final Answer:
The circumcenter is (-1, -1).