First slide

Applications of differential equations

Question

$The orthogonal trajectories of the family of ellipses x^{2} + 2 y^{2} - y = c are family of parabola$ $y = c x^{2} + k, where k =$

Moderate

A

0
B

$\frac{1}{2}$
C

$\frac{1}{3}$
D

$\frac{1}{4}$

Solution

$\begin{array}{l} x^{2} + 2 y^{2} - y = c \\ 2 x + 4 y y_{1} - y_{1} = 0 \Rightarrow 2 x = - (4 y - 1) y_{1} \\ Replace y_{1} by - \frac{\underline{1}}{\frac{d y}{d x}} \\ \Rightarrow 2 x \frac{d y}{d x} = 4 y - 1 \\ \Rightarrow \int \frac{4 d y}{4 y - 1} = \int \frac{2 d x}{x} \\ \Rightarrow 4 y - 1 = c_{1} x^{2} \\ y = c x^{2} + \frac{1}{4} k = \frac{1}{4} \end{array}$

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