If the straight line (a- 1)x - by + 4 =0 is normal to the hyperbola xy = 1 then which of the following does not hold?

Options

A. `a gt 1 ,bgt0`
B. `a gt 1, b lt 0`
C. `a lt 1, b lt 0`
D. ` a lt 1 , b gt 0`

Answer

To solve the problem, we need to analyze the conditions under which the given straight line is normal to the hyperbola \(xy = 1\).

### Step-by-Step Solution:

1. **Identify the Hyperbola and the Line**:
The hyperbola is given by the equation \(xy = 1\). The straight line is given by the equation \((a - 1)x - by + 4 = 0\).

2. **Find the Slope of the Tangent to the Hyperbola**:
To find the slope of the tangent to the hyperbola, we first differentiate \(xy = 1\) with respect to \(x\):
\[ \frac{d}{dx}(xy) = \frac{d}{dx}(1) \implies y + x\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{y}{x} \]
Thus, the slope of the tangent at the point \((x_1, y_1)\) is:
\[ m_t = -\frac{y_1}{x_1} \]

3. **Find the Slope of the Normal**:
The slope of the normal line is the negative reciprocal of the slope of the tangent:
\[ m_n = \frac{x_1}{y_1} \]

4. **Substituting into the Line Equation**:
The slope of the given line \((a - 1)x - by + 4 = 0\) can be rewritten in slope-intercept form:
\[ by = (a - 1)x + 4 \implies y = \frac{(a - 1)}{b}x + \frac{4}{b} \]
Therefore, the slope of the line is:
\[ m_l = -\frac{(a - 1)}{b} \]

5. **Setting the Slopes Equal**:
Since the line is normal to the hyperbola, we have:
\[ m_n = m_l \implies \frac{x_1}{y_1} = -\frac{(a - 1)}{b} \]

6. **Using the Condition of the Hyperbola**:
Since \((x_1, y_1)\) lies on the hyperbola, we know:
\[ x_1y_1 = 1 \implies y_1 = \frac{1}{x_1} \]
Substituting \(y_1\) into the slope equation gives:
\[ \frac{x_1}{\frac{1}{x_1}} = -\frac{(a - 1)}{b} \implies x_1^2 = -\frac{(a - 1)}{b} \]

7. **Analyzing the Conditions**:
Since \(x_1^2\) is always non-negative, the right side must also be non-negative:
\[ -\frac{(a - 1)}{b} \geq 0 \]
This implies two cases:
- Case 1: \(a - 1 < 0\) (i.e., \(a < 1\)) and \(b > 0\)
- Case 2: \(a - 1 > 0\) (i.e., \(a > 1\)) and \(b < 0\)

8. **Conclusion**:
From the above analysis, we can conclude that the following conditions must hold:
- If \(a < 1\), then \(b > 0\).
- If \(a > 1\), then \(b < 0\).

We need to identify which of the given options does not hold based on these conditions.