If the portion of the line ax - by - 1 = 0 , intercepted between the lines ax + y + 1 = 0 and x + by = 0 subtends a right angle at the origin , then find value of `4a +b^2 + (b+1)^2`.

Answer

To solve the problem, we need to find the value of \(4a + b^2 + (b+1)^2\) given that the portion of the line \(ax - by - 1 = 0\) intercepted between the lines \(ax + y + 1 = 0\) and \(x + by = 0\) subtends a right angle at the origin.

### Step 1: Find the intersection points of the lines

We have two lines:
1. \(ax + y + 1 = 0\) (let's call this Line 1)
2. \(x + by = 0\) (let's call this Line 2)

To find the intersection point of these two lines, we can solve them simultaneously.

From Line 2, we can express \(y\) in terms of \(x\):
\[ y = -\frac{x}{b} \]

Substituting this into Line 1:
\[ ax - \frac{x}{b} + 1 = 0 \]
Multiplying through by \(b\) to eliminate the fraction:
\[ abx - x + b = 0 \]
Factoring out \(x\):
\[ (ab - 1)x + b = 0 \]
Thus, we have:
\[ x = -\frac{b}{ab - 1}, \quad \text{if } ab \neq 1 \]

Now substituting \(x\) back into the equation for \(y\):
\[ y = -\frac{-\frac{b}{ab - 1}}{b} = \frac{1}{a - \frac{1}{b}} = \frac{b}{ab - 1} \]

So the intersection point \(A\) is:
\[ A\left(-\frac{b}{ab - 1}, \frac{b}{ab - 1}\right) \]

### Step 2: Find the slope of the lines

Next, we need to find the slopes of the lines to check the condition of them subtending a right angle at the origin.

**Slope of Line 1**: From \(ax + y + 1 = 0\), we can express it in slope-intercept form:
\[ y = -ax - 1 \quad \Rightarrow \quad \text{slope } m_1 = -a \]

**Slope of Line 2**: From \(x + by = 0\), we can express it as:
\[ y = -\frac{1}{b}x \quad \Rightarrow \quad \text{slope } m_2 = -\frac{1}{b} \]

### Step 3: Use the condition for perpendicular lines

For the lines to be perpendicular, the product of their slopes must equal -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes:
\[ (-a) \cdot \left(-\frac{1}{b}\right) = 1 \quad \Rightarrow \quad \frac{a}{b} = 1 \quad \Rightarrow \quad a = b \]

### Step 4: Substitute \(a = b\) into the expression

Now we need to find \(4a + b^2 + (b + 1)^2\):
Substituting \(a = b\):
\[ 4b + b^2 + (b + 1)^2 \]
Expanding \((b + 1)^2\):
\[ (b + 1)^2 = b^2 + 2b + 1 \]
Thus, the expression becomes:
\[ 4b + b^2 + b^2 + 2b + 1 = 2b^2 + 6b + 1 \]

### Step 5: Final expression

So the final value we need to evaluate is:
\[ 2b^2 + 6b + 1 \]

### Conclusion

The value of \(4a + b^2 + (b + 1)^2\) is \(2b^2 + 6b + 1\).