Question

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. are the given families of curves orthogonal trajectories of each other? that is, is every curve in one family orthogonal to every curve in the other family? x2 + y2 = ax x2 + y2 = by

Answer 1

Hello,

$P(x,y)=x^2+y^2-ax=0\\\\ \dfrac{\partial{P}}{\partial{x}}=2x-a\\\\ \dfrac{\partial{P}}{\partial{y}}=2y\\\\ Q(x,y)=x^2+y^2-by=0\\\\ \dfrac{\partial{Q}}{\partial{x}}=2x\\\\ \dfrac{\partial{Q}}{\partial{y}}=2y-b\\\\ \dfrac{\partial{P}}{\partial{x}}*\dfrac{\partial{Q}}{\partial{x}}+\dfrac{\partial{P}}{\partial{y}}*\dfrac{\partial{Q}}{\partial{y}}=(2x-a)*2x+2y(2y-b)\\ =4x^2+4y^2-2ax-2by\\ =2(2x^2+2y^2-ax-by)\\ =2*P(x,y)+Q(x,y))\\ =2*(0+0)=0\\\\ Answer YES$