Question

Given below are two statements:
Assertion (A): If a curve cut every member of a given family of curves at right angle, it is called orthogonal trajectory.
Reason (R): For the orthogonal trajectory of a differential equation, the derivative \(\dfrac{dy}{dx}\) is replaced by \(-\dfrac{dx}{dy}\).

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

For orthogonal trajectories, replace slope \(m\) by \(-\frac{1}{m}\).

Answer 1

The Correct Option is A

Concept:
Orthogonal trajectories are curves that intersect a given family of curves at right angles.

Step 1: Check Assertion.
If a curve cuts every member of a family of curves at \(90^\circ\), it is called an orthogonal trajectory. \[ A \text{ is correct} \]

Step 2: Check Reason.
If the slope of a family of curves is: \[ \frac{dy}{dx} \] then the slope of the orthogonal trajectory is the negative reciprocal: \[ -\frac{dx}{dy} \] So Reason is correct. \[ R \text{ is correct} \]

Step 3: Check explanation.
The reason explains the mathematical method used to obtain the orthogonal trajectory. \[ R \text{ correctly explains } A \] \[ \therefore \text{Correct Answer is (A)} \]