Question

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

Options

• `y((dy)/(dx))`

• `x[y  (d^2y)/(dx^2) + ((dy)/(dx))^2] - y((dy)/(dx))`

• `y^2[(d^2y)/(dx^2)] - x ((dy)/(dx))`

• None of these

Answer 1

`x[y  (d^2y)/(dx^2) + ((dy)/(dx))^2] - y((dy)/(dx))`

Explanation:

The equation of a hyperbola family with the origin as the centre and foci along the x-axis is

`x^2/a^2 - y^2/b^2` = 1  ........(1)

Differentiating w.r.t, x

`(2x)/a^2 - (2xyy^2)/b^2` = 0 or `x/a^2 - (yy^2)/b^2` = 0  ......(2)

Again differentiating

`1/a^2 - 1/b^2 (y^2 + yy^11)` = 0  ......(3)

Putting value of `1/a^2` in (2)

`1/b^2 (y^12 + yy^11) - (yy^1)/b^2` = 0

or `x(yy^11 + y^12) - yy^1` = 0

Required differential equation is `x[y  (d^2y)/(dx^2) + ((dy)/(dx))^2] - y((dy)/(dx))` = 0