Orthogonal

Calculus Level 4

Find the orthogonal trajectories to the family of curve y = c x k y=cx^k

x 2 + c y 2 = const. x^2+cy^2=\text{const.} None of these \text{ None of these } x 2 + k y 2 = const. x^2+ky^2=\text{const.} k x 2 + y 2 = const. kx^2+y^2=\text{const.} x 2 − k y 2 = const. x^2-ky^2=\text{const.}

Sabhrant Sachan by Sabhrant Sachan , 21, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Sabhrant Sachan
Nov 20, 2016

y = c x k d y d x = c k x k − 1 d y d x = k ( c x k ) x ⟹ d y d x = k y x To find orthogonal curves , we need to solve : − d x d y = k y x − ∫ x d x = k ∫ y d y − x 2 2 + c = k y 2 2 x 2 + k y 2 = const. y=cx^k \\ \dfrac{dy}{dx} = ckx^{k-1} \\ \dfrac{dy}{dx} = \dfrac{k(cx^k)}{x} \implies \dfrac{dy}{dx} = \dfrac{ky}{x} \\ \text{To find orthogonal curves , we need to solve :} \\ -\dfrac{dx}{dy} = \dfrac{ky}{x} \\ -\int xdx = k \int ydy \\ -\dfrac{x^2}{2}+c=\dfrac{ky^2}{2} \\ \boxed{x^2+ky^2 = \text{ const. }}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Your correct answer choice matches that of Choice 'D' and not 'A' (which is how I solved this particular problem). May want to consider correcting your answer key for the longer term.

tom engelsman - 4 years, 6 months ago

Log in to reply

Please report this problem, so that everyone ( including the admin ) know that there is a problem. Sorry for the inconvenience.

Sabhrant Sachan - 4 years, 6 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...