From a calculus textbook I have the following problem: show that the given curves are orthogonal:
$x^2-y^2=5 \\ 4x^2 + 9y^2 = 72$
Solving the simultaneous equation shows that the two curves meet at $(\pm3,\pm2)$, i.e. in four places. Using implicit differentiation I determine that the derivates of the two curves are $x/y$ and $-4x/9y$ respectively, and that they are orthogonal at (3,2).
So far, so good. In the worked solution in my textbook, however, it states at this point, "by symmetry, the curves are also orthogonal at (3,-2), (-3,2) and (-3,-2)". Fair enough, the two curves are symmetrical about the x-axis and so
My first question is: as a calculus student have I got to be constantly on the lookout for symmetric equations to allow me to properly address this kind of question?
My second question is: why is this observation true? Is it because:
- Both curves are symmetrical around the x-axis. Therefore if the curves are orthogonal at (3,2), they must be orthogonal at (3,-2) because the derivatives will simply have swapped signs and thus remain negative reciprocals of each other.
- Both curves are symmetrical around the y-axis. Therefore if the curves are orthogonal at (3,2) and (3,-2) they must also be orthogonal at (-3,2) and (-3,-2), because the derivatives of the two curves at those points will again just have swapped signs.