Question

If ax + by + c = 0 is normal to xy = 1, then determine if a and b are less than, greater than, or equal to zero.

Answer 1

To determine the relationship between a and b, we need to find the slope of the line ax + by + c = 0 and compare it to the slope of the line xy = 1.

The line ax + by + c = 0 can be rewritten as y = (-a/b)x - c/b, where the slope of this line is -a/b.

The line xy = 1 can be rewritten as y = 1/x, which can also be written as y = x^(-1). The slope of this line is -1.

For the line ax + by + c = 0 to be normal to xy = 1, the product of their slopes should be -1:

(-a/b) * (-1) = 1

Simplifying the equation:

a/b = 1

This tells us that a and b must have the same sign. If both a and b are positive or negative, the product will be positive, not -1. Therefore, a and b must be either both positive or both negative.

In conclusion, a and b are either both greater than zero or both less than zero.