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Question

(A) The equation of the family of hyperbolas with centres at the origin and axes along the coordinate axes is given by $\frac{x^2}{a^2} - \frac{y^2}{b

Vedclass · Accepted Answer

$x y y_2 + x y_1^2 - y y_1 = 0$

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(A) The equation of the family of hyperbolas with centres at the origin and axes along the coordinate axes is given by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (or $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ for the general conic form). Differentiating with respect to $x$: $\frac{2x}{a^2} - \frac{2y}{b^2} y_1 = 0 \Rightarrow \frac{x}{a^2} = \frac{y y_1}{b^2} \Rightarrow \frac{y y_1}{x} = \frac{b^2}{a^2} = k$ (constant).Differentiating again with respect to $x$: $\frac{d}{dx} \left( \frac{y y_1}{x} ight) = 0$.Using the quotient rule: $\frac{x(y y_2 + y_1^2) - y y_1}{x^2} = 0$.Since $x 
eq 0$,we have $x y y_2 + x y_1^2 - y y_1 = 0$.

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

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