The opposite reciprocal of 9 is -\frac{1}{9}.
To find this, you first take the reciprocal of 9, which is \frac{1}{9}. The reciprocal of a number is simply 1 divided by that number. Then, to find the opposite of the reciprocal, you change its sign. Since the reciprocal of 9 is positive, the opposite reciprocal is negative, resulting in -\frac{1}{9}. This concept is often used in mathematics, particularly in algebra when dealing with slopes of perpendicular lines.