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: Understanding the Concept:
The pair of lines equation can be factorized into two linear lines.
The triangle's nature (right-angled, equilateral, etc.) determines the circumradius easily.
Step 2: Key Formula or Approach:
$xy + 2x + 2y + 4 = 0 \implies (x+2)(y+2) = 0$.
Lines are $x = -2$ and $y = -2$.
Third line is $x + y + 2 = 0$.
Step 3: Detailed Explanation:
Vertices are $(-2, -2), (-2, 0), (0, -2)$.
It is a right-angled triangle at $(-2, -2)$ with legs of length 2.
Hypotenuse $c = \sqrt{2^2 + 2^2} = 2\sqrt{2}$.
Circumradius $R = c/2 = \sqrt{2}$.
Step 4: Final Answer:
The circumradius is $\sqrt{2}$ units.