Question

In \( \mathbb{R}^2 \), the family of trajectories orthogonal to the family of asteroids \( x^{2/3} + y^{2/3} = a^{2/3} \) is given by

• A. \( x^{4/3} + y^{4/3} = c^{4/3} \)
• B. \( x^{4/3} - y^{4/3} = c^{4/3} \)
• C. \( x^{5/3} - y^{5/3} = c^{5/3} \)
• D. \( x^{2/3} - y^{2/3} = c^{2/3} \)

Hint:

To find the orthogonal trajectories of a given family of curves, differentiate implicitly and use the orthogonality condition that the slopes of the curves must multiply to \( -1 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Problem.
The family of asteroids is given by \( x^{2/3} + y^{2/3} = a^{2/3} \), and we are looking for a family of curves orthogonal to these curves. The key property of orthogonal trajectories is that the product of their slopes must be \( -1 \).

Step 2: Deriving the Orthogonal Family.
To find the family of curves orthogonal to the given family of asteroids, we take the derivative of \( x^{2/3} + y^{2/3} = a^{2/3} \) implicitly and use the fact that the slopes of the curves must satisfy the orthogonality condition. The correct form of the orthogonal trajectory is \( x^{4/3} + y^{4/3} = c^{4/3} \).

Step 3: Conclusion.
The correct answer is (A) \( x^{4/3} + y^{4/3} = c^{4/3} \).