Consider a triangle ABC with vertex `A(2, -4)`. The internal bisectors of the angle B and C are `x+y=2` and`x- 3y = 6` respectively. Let the two bisectors meet at `I`.if (a, b) is incentre of the triangle ABC then `(a + b)` has the value equal to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the given information
We have a triangle ABC with vertex A at the coordinates (2, -4). The internal angle bisectors of angles B and C are given by the equations:
1. \( x + y = 2 \) (for angle B)
2. \( x - 3y = 6 \) (for angle C)

### Step 2: Find the intersection point of the two angle bisectors
To find the coordinates of point I, which is the intersection of the two lines, we will solve the equations simultaneously.

1. From the first equation \( x + y = 2 \), we can express \( x \) in terms of \( y \):
\[ x = 2 - y \]

2. Substitute \( x \) in the second equation \( x - 3y = 6 \):
\[ (2 - y) - 3y = 6 \]
Simplifying this gives:
\[ 2 - 4y = 6 \]
\[ -4y = 6 - 2 \]
\[ -4y = 4 \]
\[ y = -1 \]

3. Now substitute \( y = -1 \) back into the first equation to find \( x \):
\[ x + (-1) = 2 \]
\[ x - 1 = 2 \]
\[ x = 3 \]

Thus, the intersection point \( I \) is \( (3, -1) \).

### Step 3: Identify the coordinates of the incenter
The coordinates of the incenter \( (a, b) \) of triangle ABC are given by the coordinates of point I, which we found to be:
- \( a = 3 \)
- \( b = -1 \)

### Step 4: Calculate \( a + b \)
Now, we need to find the value of \( a + b \):
\[ a + b = 3 + (-1) = 3 - 1 = 2 \]

### Final Answer
The value of \( a + b \) is \( 2 \).

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