If the line `ax+by+c=0` is normal to the curve `xy+5=0`, then

Options

A. `a gt0, b gt0`
B. `b gt 0, a lt 0`
C. `blt0, a gt0`
D. none of these

Answer

To solve the problem, we need to find the conditions under which the line \( ax + by + c = 0 \) is normal to the curve defined by \( xy + 5 = 0 \).

### Step 1: Understand the curve
The given curve is \( xy + 5 = 0 \). We can rewrite this as \( xy = -5 \). This represents a hyperbola.

### Step 2: Differentiate the curve
To find the slope of the tangent to the curve, we differentiate \( xy + 5 = 0 \) implicitly with respect to \( x \):
\[ \frac{d}{dx}(xy) + \frac{d}{dx}(5) = 0 \]
Using the product rule on \( xy \):
\[ y + x\frac{dy}{dx} = 0 \]
Rearranging gives:
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 3: Find the slope of the normal
The slope of the normal line is the negative reciprocal of the slope of the tangent. Therefore, if the slope of the tangent is \( -\frac{y}{x} \), the slope of the normal is:
\[ \text{slope of normal} = \frac{x}{y} \]

### Step 4: Relate the normal's slope to the line equation
The equation of the line is \( ax + by + c = 0 \). We can rearrange this to find the slope:
\[ by = -ax - c \implies y = -\frac{a}{b}x - \frac{c}{b} \]
Thus, the slope of this line is \( -\frac{a}{b} \).

### Step 5: Set the slopes equal
Since the line is normal to the curve, we set the slopes equal:
\[ \frac{x}{y} = -\frac{a}{b} \]
This implies:
\[ x \cdot b = -a \cdot y \]

### Step 6: Substitute for \( xy \)
From the curve equation \( xy = -5 \), we can express \( y \) in terms of \( x \):
\[ y = -\frac{5}{x} \]
Substituting this into the equation \( x \cdot b = -a \cdot y \):
\[ x \cdot b = -a \left(-\frac{5}{x}\right) \implies bx^2 = 5a \]

### Step 7: Analyze the signs
From \( bx^2 = 5a \), we can analyze the signs:
- If \( b > 0 \), then \( a \) must also be positive for the equation to hold true.
- If \( b < 0 \), then \( a \) must also be negative.

Thus, we conclude that \( \frac{a}{b} > 0 \).

### Conclusion
The line \( ax + by + c = 0 \) is normal to the curve \( xy + 5 = 0 \) if \( a \) and \( b \) have the same sign. Therefore, we can summarize the conditions as:
- \( a > 0 \) and \( b > 0 \) or
- \( a < 0 \) and \( b < 0 \).