Hi, how do I find the orthogonal trajectories of the family of ellipses: $\displaystyle \frac{x^2}{a^2} + y^2 = k^2$ where a is a constant? Thanks.
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Originally Posted by coldfire Hi, how do I find the orthogonal trajectories of the family of ellipses: $\displaystyle \frac{x^2}{a^2} + y^2 = k^2$ where a is a constant? Thanks. First calculate the slope of the tangent $\displaystyle \frac{2x}{a^2} + 2 y \frac{dy}{dx} = 0 \;\text{or}\; \frac{dy}{dx} = - \frac{x}{a^2 y}. $ Orthogonal trajectories will be perpendicular so $\displaystyle \frac{dy}{dx} = \frac{a^2y}{x}$. Now integrate.
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