Statement -1 : The circumcentre of the triangle formed by the lines x + y = 0 , x - y = 0 and x + 5 = 0 ( - 5 , 0) .
Statement-2 : Cicumcentre of a triangle lies inside the triangle

Options

A. Statement -1 is True , Statement - 2 is true , Statement- 2 is a correct explanation for statement - 5
B. Statement-1 is True , Statement-2 is True , Statement -2 is not a correct explanation for Statement - 1 .
C. Statement-1 is True , Statement - 2 is False .
D. Statement - 1 is False , Statement -2 is True .

Explanation

To solve the problem, we need to analyze the statements regarding the circumcenter of the triangle formed by the given lines. Let's go through the steps systematically.

### Step 1: Identify the Lines
The lines given are:
1. \( x + y = 0 \) (Line 1)
2. \( x - y = 0 \) (Line 2)
3. \( x + 5 = 0 \) (Line 3)

### Step 2: Find the Points of Intersection
To form a triangle, we need to find the points of intersection of these lines.

1. **Intersection of Line 1 and Line 2:**
\[ x + y = 0 \quad \text{(1)} \]
\[ x - y = 0 \quad \text{(2)} \]
From (2), we have \( y = x \). Substituting into (1):
\[ x + x = 0 \implies 2x = 0 \implies x = 0 \implies y = 0 \]
So, the intersection point is \( (0, 0) \).

2. **Intersection of Line 1 and Line 3:**
\[ x + y = 0 \quad \text{(1)} \]
\[ x + 5 = 0 \quad \text{(3)} \]
From (3), we have \( x = -5 \). Substituting into (1):
\[ -5 + y = 0 \implies y = 5 \]
So, the intersection point is \( (-5, 5) \).

3. **Intersection of Line 2 and Line 3:**
\[ x - y = 0 \quad \text{(2)} \]
\[ x + 5 = 0 \quad \text{(3)} \]
From (3), we have \( x = -5 \). Substituting into (2):
\[ -5 - y = 0 \implies y = -5 \]
So, the intersection point is \( (-5, -5) \).

### Step 3: Identify the Vertices of the Triangle
The vertices of the triangle formed by the lines are:
1. \( A(0, 0) \)
2. \( B(-5, 5) \)
3. \( C(-5, -5) \)

### Step 4: Determine the Circumcenter
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. In this case, since two of the lines are perpendicular (Line 1 and Line 2), the circumcenter will be located at the midpoint of the hypotenuse.

1. **Find the midpoint of side \( BC \):**
\[ B(-5, 5) \quad C(-5, -5) \]
Midpoint \( M \) is given by:
\[ M = \left( \frac{-5 + (-5)}{2}, \frac{5 + (-5)}{2} \right) = \left( -5, 0 \right) \]

### Step 5: Analyze the Statements
- **Statement 1:** The circumcenter of the triangle formed by the lines is \( (-5, 0) \). This is **true**.
- **Statement 2:** The circumcenter of a triangle lies inside the triangle. In this case, the circumcenter \( (-5, 0) \) lies on the side of the triangle, not inside. This is **false**.

### Conclusion
- **Final Answer:** Statement 1 is true, and Statement 2 is false.

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