Chapter 6.4, Problem 86E

To determine

Whether the statement “If the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a,b>0$ intersect at right angles, then $a=b$ .” is true or false.

Answer to Problem 86E

The statement “If the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a,b>0$ intersect at right angles, then $a=b$ .” is true.

Explanation of Solution

Given information:

The statement “If the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a,b>0$ intersect at right angles, then $a=b$ .”

Consider the provided statement “If the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a,b>0$ intersect at right angles, then $a=b$ .”

The statement is true because horizontal asymptotes for a curve is the limit value of the function when x tends to positive infinity and negative infinity.

For the provided hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ it is observed that it is a horizontal hyperbola.

The slopes of asymptotes of horizontal hyperbolas are $\pm \frac{b}{a}$ .

Two curves intersect at right angles if product of their slope is $-1$ .

That is $\pm \frac{b}{a}=\pm 1$ . This condition is only possible when $a=b$ .

Thus, the statement “If the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a,b>0$ intersect at right angles, then $a=b$ .” is true.