Question

# Consider the family of circles$x ^ { 2 } + y ^ { 2 } = 2 c x.$Show that the differential equation for determining the family of orthogonal trajectories is$\frac { d y } { d x } = \frac { 2 x y } { x ^ { 2 } - y ^ { 2 } }.$

Solution

Verified
Step 1
1 of 6

$x^2+y^2=2cx\rightarrow 2x+2y\dfrac{dy}{dx}=2c$

We are given the original family of curves and their orthogonal trajectories. The goal is to find the orthogonal trajectory of the given equation and show that it is the same as the given orthogonal trajectory. So first, take the given equation and derive implicitly with respect to x.

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