Question

If the line \[ ax+by+c=0 \] is a normal to the curve \[ xy=1, \] then:

• A. \(a\gt 0,\ b\gt 0\)
• B. \(a\gt 0,\ b\lt 0\)
• C. \(a\gt 0,\ b=0\)
• D. \(a\lt 0,\ b\lt 0\)

Hint:

For a line \[ ax+by+c=0, \] the slope is \[ -\frac{a}{b}. \] Use the sign of the slope to determine the signs of \(a\) and \(b\).

Answer 1

The Correct Option is B

Step 1: Find the slope of tangent to the curve.
Given curve: \[ xy=1 \] Differentiate implicitly: \[ x\frac{dy}{dx}+y=0 \] \[ \frac{dy}{dx}=-\frac{y}{x} \] Thus, slope of tangent is \[ m_t=-\frac{y}{x} \]

Step 2: Find the slope of the normal.
Slope of normal is the negative reciprocal of tangent slope: \[ m_n=\frac{x}{y} \] Since \[ xy=1, \] we have \[ y=\frac{1}{x} \] Therefore, \[ m_n=\frac{x}{1/x} \] \[ m_n=x^2 \] Hence, \[ m_n\gt 0 \]

Step 3: Compare with the slope of the line.
For the line \[ ax+by+c=0, \] the slope is \[ -\frac{a}{b} \] Since the line is a normal, \[ -\frac{a}{b}\gt 0 \] Thus, \(a\) and \(b\) must have opposite signs.
Among the given options, only \[ a\gt 0,\quad b\lt 0 \] satisfies this condition.

Step 4: Final conclusion.
Hence, \[ \boxed{a\gt 0,\ b\lt 0} \]