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 Question:
We are given a pair of lines combined into one quadratic equation and a third independent line. We must find the circumcenter of the resulting triangle.

Step 2: Detailed Explanation:
First, factor the joint equation of the two lines:
$xy + 2x + 2y + 4 = 0$
Group the terms:
$x(y + 2) + 2(y + 2) = 0$
$(x + 2)(y + 2) = 0$
This gives us two distinct perpendicular straight lines:
Line 1: $x = -2$ (A vertical line)
Line 2: $y = -2$ (A horizontal line)
The third line is given as:
Line 3: $x + y + 2 = 0$
Because a vertical line and a horizontal line intersect at exactly $90^\circ$, the triangle formed by these three lines is fundamentally a right-angled triangle.
The right-angle vertex is located exactly at the intersection of Line 1 and Line 2: $(-2, -2)$.
A profound property of right-angled triangles in geometry is that their circumcenter is exactly the midpoint of their hypotenuse.
The hypotenuse is the segment of Line 3 bounded by its intersections with Line 1 and Line 2.
Let's find the endpoints of the hypotenuse:
Intersection of Line 1 ($x = -2$) and Line 3 ($x + y + 2 = 0$):
$(-2) + y + 2 = 0 \implies y = 0$. Point A = $(-2, 0)$.
Intersection of Line 2 ($y = -2$) and Line 3 ($x + y + 2 = 0$):
$x + (-2) + 2 = 0 \implies x = 0$. Point B = $(0, -2)$.
The circumcenter is the midpoint of A$(-2, 0)$ and B$(0, -2)$:
$\text{Midpoint} = \left( \frac{-2 + 0}{2}, \frac{0 + (-2)}{2} \right)$
$\text{Midpoint} = (-1, -1)$.

Step 3: Final Answer:
The circumcenter is (-1, -1), matching option (c).