If the line `ax+by+c=0` is normal to the `xy+5=0`, then a and b have

Options

A. same sign
B. opposite sign
C. cannot be discussed
D. None of these

Answer

To solve the problem, we need to determine the relationship between the coefficients \( a \) and \( b \) of the line \( ax + by + c = 0 \) that is normal to the line defined by \( xy + 5 = 0 \).

### Step-by-Step Solution:

1. **Identify the Given Line**:
The line given is \( xy + 5 = 0 \). We can rewrite this as \( y = -\frac{5}{x} \).

2. **Differentiate to Find the Slope**:
To find the slope of the tangent to the curve, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(-\frac{5}{x}) = \frac{5}{x^2} \]
The slope of the tangent line at any point \( (x_1, y_1) \) is:
\[ m_t = \frac{5}{x_1^2} \]

3. **Find the Slope of the Normal**:
The slope of the normal line, which is perpendicular to the tangent, is given by:
\[ m_n = -\frac{1}{m_t} = -\frac{x_1^2}{5} \]

4. **Equation of the Normal Line**:
The equation of the normal line at the point \( (x_1, y_1) \) can be expressed as:
\[ y - y_1 = m_n (x - x_1) \]
Substituting \( m_n \):
\[ y - y_1 = -\frac{x_1^2}{5}(x - x_1) \]
Rearranging gives:
\[ 5(y - y_1) = -x_1^2(x - x_1) \]
Expanding this:
\[ 5y - 5y_1 = -x_1^2x + x_1^3 \]
Rearranging leads to:
\[ x_1^2x + 5y - x_1^3 - 5y_1 = 0 \]

5. **Comparing with the Given Line**:
The normal line can be compared to the form \( ax + by + c = 0 \). Thus, we identify:
\[ a = x_1^2, \quad b = 5, \quad c = -x_1^3 - 5y_1 \]

6. **Condition from the Original Equation**:
Since \( (x_1, y_1) \) lies on the curve \( xy + 5 = 0 \), we have:
\[ x_1y_1 + 5 = 0 \implies y_1 = -\frac{5}{x_1} \]

7. **Substituting \( y_1 \)**:
Substitute \( y_1 \) into the expression for \( b \):
\[ b = 5 \implies b = 5 \]
Now substituting \( y_1 \) into \( c \):
\[ c = -x_1^3 - 5\left(-\frac{5}{x_1}\right) = -x_1^3 + \frac{25}{x_1} \]

8. **Finding the Relationship Between \( a \) and \( b \)**:
Since \( a = x_1^2 \) and \( b = 5 \), we can conclude:
- If \( x_1 > 0 \), then \( a > 0 \).
- If \( x_1 < 0 \), then \( a < 0 \).

Therefore, \( a \) and \( b \) will have the same sign.

### Conclusion:
The coefficients \( a \) and \( b \) have the same sign.