Orthogonal family of curves to the level lines of $f(x,y)=xy-1$

Question

Find the orthogonal family of curves to the level lines of $f(x,y)=xy-1$.

---

My attemp:

The level lines are $$\left\lbrace(x,y)\in\mathbb R^2:xy-1=K\quad\text{for some constant }K\right\rbrace\text.$$

Differentiating both sides: $$\mathfrak F:y+xy'=0\Rightarrow\mathfrak F^\perp:y-\frac x{y'}=0\Rightarrow\mathfrak F^\perp:\frac{y^2}2-\frac{x^2}2=c\Rightarrow\boxed{\mathfrak F^\perp:y^2-x^2=C,\;C\in\mathbb R}\text.$$

Is that correct?

Thanks!

Answer 1

Yes, your answer is correct.

You can also verify your result by graphing those curves and see the level curves are orthogonal to each other at every point of intersection.

Answer 2

Yes it is correct indeed we have

• $xy-1=K \implies ydx+xdy=0 \quad m_1=\frac{dy}{dx}=-\frac{y}{x}$

• $\frac{y^2}2-\frac{x^2}2=c \implies ydy-xdx=0 \quad m_2=\frac{dy}{dx}=\frac{x}{y}$

and $$m_1\cdot m_2=-\frac{y}{x}\cdot \frac{x}{y}=-1$$