Advertisements
Advertisements
Question
Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.
Advertisements
Solution
The equation of the hyperbola is
`"x"^2/16 - "y"^2/36 = "k"` i.e. `"x"^2/"16 k" - "y"^2/"36k" = 1`
Comparing this equation with `"x"^2/"a"^2 - "y"^2/"b"^2 = 1`, we get
a2 = 16k, b2 = 36k
∴ a = `4sqrt"k", "b" = 6sqrt"k"`
∴ l(transverse axis) = 2a = `8sqrt"k"`
and l(conjugate axis) = 2b = `12sqrt"k"`
Let 2A and 2B be the lengths of the transverse and conjugate axes of the required hyperbola.
Then according to the given condition
2A = a = `4sqrt"k" and 2"B" = "b" = 6sqrt"k"`
∴ A = `2sqrt"k"` and B = `3sqrt"k"`
∴ equation of the required hyperbola is
`"x"^2/"A"^2 - "y"^2/"B"^2 = 1`
i.e. `"x"^2/"4k" - "y"^2/"9k" = 1`
∴ 9x2 - 4y2 = 36k, where k is an arbitrary constant.
Differentiating w.r.t. x, we get
`9 xx "2x" - 4 xx "2y" "dy"/"dx" = 0`
∴ `"9x" - "4y" "dy"/"dx" = 0`
This is the required D.E.
APPEARS IN
RELATED QUESTIONS
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
x3 + y3 = 4ax
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = A cos (log x) + B sin (log x)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a + `"a"/"x"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = e−2x (A cos x + B sin x)
Find the differential equation of the ellipse whose major axis is twice its minor axis.
In the following example verify that the given expression is a solution of the corresponding differential equation:
xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`
The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` is ______.
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
In the following example verify that the given function is a solution of the differential equation.
`"x"^2 = "2y"^2 log "y", "x"^2 + "y"^2 = "xy" "dx"/"dy"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
The general solution of `(dy)/(dx)` = e−x is ______.
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
Form the differential equation of y = (c1 + c2)ex
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
If `x^2 y^2 = sin^-1 sqrt(x^2 + y^2) + cos^-1 sqrt(x^2 + y^2)`, then `"dy"/"dx"` = ?
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
For the curve C: (x2 + y2 – 3) + (x2 – y2 – 1)5 = 0, the value of 3y' – y3 y", at the point (α, α), α < 0, on C, is equal to ______.
The differential equation of all circles passing through the origin and having their centres on the X-axis is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
The differential equation for a2y = log x + b, is ______.
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
