Question

The differential equation of the family of hyperbolas having their centers at origin and their axes along the coordinate axes is:

• A. \( xy y_2 + xy_1^2 - yy_1 = 0 \)
• B. \( xy_2 - xy y_1^2 + yy_1 = 0 \)
• C. \( xy_2 + xy_1^2 + yy_1 = 0 \)
• D. \( xy_2 + xy_1^2 - yy_1 = 0 \)

Hint:

For obtaining the differential equation of a family of curves, differentiate the given equation repeatedly and eliminate any arbitrary constants or parameters.

Answer 1

The Correct Option is A

Step 1: General Equation of Hyperbola
The general equation of a hyperbola centered at the origin with axes along the coordinate axes is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \] This represents a family of hyperbolas where \( a \) and \( b \) are parameters. Step 2: Eliminating Parameters
To obtain the differential equation, we differentiate the given equation with respect to \( x \): \[ \frac{2x}{a^2} - \frac{2yy_1}{b^2} = 0. \] Differentiating again: \[ \frac{2}{a^2} - \frac{2(y_1^2 + yy_2)}{b^2} = 0. \] Multiplying throughout by \( b^2 \) and simplifying: \[ xy y_2 + xy_1^2 - yy_1 = 0. \] Step 3: Conclusion
Thus, the required differential equation of the family of hyperbolas is: \[ \boxed{xy y_2 + xy_1^2 - yy_1 = 0.} \]