Question

If the line ax+by+c=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

Answer 1

Answer: (B)

a>0, b<0

Answer 2

The curve xy = 1 can be written as y = 1/x, which means that the derivative of y with respect to x is: dy/dx = -1/x^2

For a normal to the curve at a given point, the slope of the tangent at that point is given by: m = -1/dy/dx = x^2

Therefore, the equation of the tangent at the point (a, 1/a) is: y - 1/a = x^2 (x - a)

Simplifying, we get: y = a^2 x + (1 - a^3)/a This is the equation of the tangent line.

For this line to be a normal to the curve xy = 1, it must be perpendicular to the curve at the point (a, 1/a).

The slope of the curve at this point is: dy/dx = -1/x^2 = -a^2

Therefore, the slope of the line perpendicular to the curve is: m = 1/a^2

This means that the product of the slopes of the tangent and the normal at the point (a, 1/a) is: m * (-a^2) = -1

Solving for a, we get: a = ±1 Substituting a = ±1 in the equation of the tangent line, we get: y = ±x + 1

These are the equations of the two lines that are normal to the curve at the points (1, 1) and (-1, -1).

The normal at (1, 1) has a positive slope, and the normal at (-1, -1) has a negative slope. Therefore, the correct option is (B) a > 0, b < 0.