Question

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

• A. 2 units
• B. 1 unit
• C. $\sqrt{2}$ units
• D. $\sqrt{3}$ units

Hint:

In a right-angled triangle, the circumradius is exactly half of the hypotenuse.

Answer 1

The Correct Option is C

Step 1: Simplify Pair of Lines
$xy + 2x + 2y + 4 = 0 \implies x(y+2) + 2(y+2) = 0 \implies (x+2)(y+2) = 0$.
The lines are $x = -2$ and $y = -2$. These are perpendicular.
Step 2: Identify Triangle
Triangle bounded by $x = -2, y = -2$ and $x + y = -2$.
Vertices are intersection of: 1. $x = -2, y = -2 \to (-2, -2)$ 2. $x = -2, x + y = -2 \to (-2, 0)$ 3. $y = -2, x + y = -2 \to (0, -2)$
Step 3: Circumradius
This is a right-angled triangle. Hypotenuse is the distance between $(-2, 0)$ and $(0, -2)$.
$Hyp = \sqrt{(2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}$.
Circumradius $R = Hyp/2 = \sqrt{2}$.
Final Answer: (C)