Question

Find the orthogonal trajectory of family of
a is parameter.
Curves y=x^3-
a3/3x
where
a is a parameter​

Answer 1

The orthogonal trajectory of the given family of curves is given by the equation |$\frac{3x^2 - a^3}{3}$| = K$e^{-x}$, where K is a constant.

Explanation:

To find the orthogonal trajectory of the given family of curves, we need to determine a new family of curves that intersect the original curves at right angles.

Let's first find the slope of the given family of curves. We differentiate the equation of the curves with respect to x:

= $\frac{dy}{dx} = 3x^2 - \frac{a^{3} }{3}$

The slope of the orthogonal trajectory at any point will be the negative reciprocal of the slope of the original curve at that point. So, the slope of the orthogonal trajectory is given by:

m = $\frac{-1}{3x^2 - \frac{a^{3} }{3}}$

To find the equation of the orthogonal trajectory, we integrate the above expression with respect to x:

∫($\frac{1}{3x^2 - \frac{a^{3} }{3}}$) dx = ∫(-dx)

Simplifying the integral, we get:

ln|$\frac{3x^2 - a^3}{3}$| = -x + C

Where C is the constant of integration.

Finally, we exponentiate both sides to eliminate the natural logarithm:

|$\frac{3x^2 - a^3}{3}$| = $e^{(-x+C)}$

Simplifying further, we have:

|$\frac{3x^2 - a^3}{3}$| = K$e^{-x}$

Where K is the constant of integration.

Thus, the orthogonal trajectory of the given family of curves is given by the equation |$\frac{3x^2 - a^3}{3}$| = K$e^{-x}$, where K is a constant.

For more such questions on orthogonal trajectory:

https://brainly.in/question/48250119

#SPJ1