Question

If the line \(ax+by+5 = 0\) is a normal to the curve \(xy = 1\) then .....

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

Hint:

Slope of the normal to xy = 1 is x squared, which is always positive.

Answer 1

The Correct Option is A

Step 1: Tangent and normal slopes:
Differentiating $xy = 1$ gives $y + xy' = 0$, so $y' = -\frac yx = -\frac{1}{x^2}$. The normal has slope $x^2$, which is positive.

Step 2: Compare with the line:
Slope of $ax + by + 5 = 0$ is $-\frac ab$. It must be positive, which means $\frac ab < 0$.

Step 3: Choose:
Only option (A), where $a$ is positive and $b$ is negative, fits this. Options where both have the same sign give $\frac ab > 0$, and $b = 0$ makes the line vertical.

Final Answer:
Option (A). \[ \boxed{a>0,\ b<0} \]