Verify that two families of curves are orthogonal where $C$ and $K$ are real numbers.

Question

Use a graphing to graph the two families for two values of $C$ and two values of $K$. $xy=C$, $x^2-y^2=K$

Here is what I got:

$xy'+y=0$ and $2x-2yy'=0$

$y'=\frac{-y}{x}$ and $y'=\frac{x}{y}$

Since $\frac{-y}{x}\cdot\frac{x}{y}=-1$, the two curves are orthogonal for some values of $C$ and some values of $K$.

How do I find those values? It says to use the calculator to graph the curve for two values of $C$ and two Values of $K$. How? Using a system?

Thanks beforehand,

Answer 1

The statements is about two families of curves, each described by a parameter ($C$ in one case, $K$ in the other). What you have shown is that wherever a curve from the first family (for any $C$) intersects a curve from the second family (for any $K$), the tangent lines to each curve are perpendicular.

So you just have two choose two values of $C$ and two values of $K$ and graph them. You might try $C=\pm1$ and $K=\pm1$.