Question

find the orthogonal trajectory of the family of curve x^2+y^2=c^2​

Answer 1

Answer:

First, we find that the family of curves satisfies the differential equation

\[ x^2 - y^2 = C \quad \implies \quad 2x - 2yy' = 0 \quad \implies \quad y' = \frac{x}{y}. \]

Therefore, the orthogonal trajectories satisfy the differential equation

\begin{align*} y' = -\frac{y}{x} && \implies && \frac{1}{y} y' &= -\frac{1}{x} \\ && \implies && \int \frac{1}{y} \, dy &= -\int \frac{1}{x} \, dx \\ && \implies && \log |y| &= -\log|x| + C \\ && \implies && y &= \frac{C}{x} \\ && \implies && xy &= C. \end{align*}