Question

If the line \( ax + by + c = 0 \) is a normal to the curve \( xy = 1 \), then which of the following is true?

• A. \( a>0, b>0 \)
• B. \( a>0, b<0 \)
• C. \( a>0, b = 0 \)
• D. \( a<0, b<0 \)

Hint:

Slope of normal line equals negative reciprocal of derivative at that point.

Answer 1

The Correct Option is B

The curve \( xy = 1 \) has the property that the derivative at any point \( (x_0, y_0) \) is: \[ \begin{align} \frac{dy}{dx} = -\frac{y}{x} \Rightarrow \text{slope of tangent} = -\frac{y_0}{x_0} \Rightarrow \text{slope of normal} = \frac{x_0}{y_0} \] Suppose line is normal: \( ax + by + c = 0 \Rightarrow \) slope = \( -\frac{a}{b} = \frac{x_0}{y_0} \Rightarrow ab<0 \)
So \( a>0 \Rightarrow b<0 \)