Question

Form the differential equation of the family of hyperbolas having foci on y-axis and centre at origin.

Answer 1

equation of hyperbola when focus on y axis

Answer 2

Answer:

$y^{2} \frac{d^{2}y }{dx^{2} }=\frac{y^{2} }{x} \frac{dy}{dx} -y(\frac{dy}{dx}) ^{2}$

Step-by-step explanation:

Equation of hyperbola having foci on y-axis and center at origin is given by, $\frac{y^{2} }{a^{2} } -\frac{x^{2} }{b^{2} } =1$ ....... (1)

Now, differentiating (1) with respect to x,

$\frac{2y}{a^{2} }. \frac{dy}{dx} =\frac{2x}{b^{2} }$

⇒  $\frac{dy}{dx}=\frac{a^{2} }{b^{2} } (\frac{x}{y} )$ ....... (2)

Again differentiating with respect to x.

⇒$\frac{d^{2}y }{dx^{2} } =\frac{a^{2} }{b^{2} } (\frac{y-x\frac{dy}{dx} }{y^{2} } )$ {From equation (2)}

⇒$y^{2} \frac{d^{2}y }{dx^{2} }=\frac{y}{x} \frac{dy}{dx}(y-x\frac{dy}{dx} )$

⇒$y^{2} \frac{d^{2}y }{dx^{2} }=\frac{y^{2} }{x} \frac{dy}{dx} -y(\frac{dy}{dx}) ^{2}$

Hence this is the required differential equation. (Answer)