The point `(a^(2),a+1)` lies in the angle between the lines `3x+y+1=0` and `x+2y-5=0` containing the origin. If a is an integer, then the sum of all possible values of a is

Options

A. `-2`
B. `-3`
C. `-1`
D. 2

Answer

To solve the problem, we need to determine the integer values of \( a \) such that the point \( (a^2, a + 1) \) lies in the angle between the lines \( 3x + y + 1 = 0 \) and \( x + 2y - 5 = 0 \) which contain the origin.

### Step 1: Identify the lines and their conditions
The lines are given by:
1. \( 3x + y + 1 = 0 \)
2. \( x + 2y - 5 = 0 \)

We need to check whether the point \( (a^2, a + 1) \) and the origin \( (0, 0) \) lie on the same side of both lines.

### Step 2: Check the first line
Substituting the origin \( (0, 0) \) into the first line:
\[ 3(0) + (0) + 1 = 1 > 0 \]
This means the origin is on the side where the expression is positive.

Now, substituting the point \( (a^2, a + 1) \):
\[ 3(a^2) + (a + 1) + 1 = 3a^2 + a + 2 \]
For the point to be on the same side as the origin, we need:
\[ 3a^2 + a + 2 > 0 \]

### Step 3: Analyze the inequality
The quadratic \( 3a^2 + a + 2 \) is always positive because its discriminant is negative:
\[ D = b^2 - 4ac = 1^2 - 4(3)(2) = 1 - 24 = -23 < 0 \]
Thus, this inequality holds for all real \( a \).

### Step 4: Check the second line
Now, substituting the origin \( (0, 0) \) into the second line:
\[ (0) + 2(0) - 5 = -5 < 0 \]
This means the origin is on the side where the expression is negative.

Now substituting the point \( (a^2, a + 1) \):
\[ (a^2) + 2(a + 1) - 5 = a^2 + 2a + 2 - 5 = a^2 + 2a - 3 \]
For the point to be on the same side as the origin, we need:
\[ a^2 + 2a - 3 < 0 \]

### Step 5: Factor the quadratic inequality
Factoring the quadratic:
\[ a^2 + 2a - 3 = (a + 3)(a - 1) < 0 \]
To find the intervals where this inequality holds, we can analyze the sign changes around the roots \( a = -3 \) and \( a = 1 \).

### Step 6: Determine the intervals
The quadratic \( (a + 3)(a - 1) < 0 \) is negative between the roots:
\[ -3 < a < 1 \]

### Step 7: Identify integer values of \( a \)
The integer values of \( a \) in the interval \( (-3, 1) \) are:
\[ -2, -1, 0 \]

### Step 8: Calculate the sum of all possible values of \( a \)
Now, we sum these integer values:
\[ -2 + (-1) + 0 = -3 \]

### Final Answer
The sum of all possible integer values of \( a \) is \( \boxed{-3} \).