Advertisements
Advertisements
प्रश्न
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
उत्तर १
Here, the equation of the given hyporbola is
x2 - 3y2 - 4x = 8
⇒ x2 - 4x + 4 - 3y2 = 8 + 4
⇒ (x - 2)2 - 3y2 = 12
⇒ `("x" - 2)^2/12 - "y"^2/4 = 1`
Here, a2 = 12 and b2 = 4 ⇒ a = `2sqrt3` and b = 2
For centre put X = 0 and Y = 0
x - 2 = 0 and y = 0
x = 2 and y = 0
∴ Coordinates of the centre are (2,0)
For foci, X = ± ae and Y = 0
Now, e = `sqrt("a"^2 + "b"^2)/"a"^2 = sqrt(12 + 4)/12 = sqrt(16/12) = 4/2sqrt3 = 2/sqrt3`
उत्तर २
Here, the equation of the given hyporbola is
x2 - 3y2 - 4x = 8
⇒ x2 - 4x + 4 - 3y2 = 8 + 4
⇒ (x - 2)2 - 3y2 = 12
⇒ `("x" - 2)^2/12 - "y"^2/4 = 1`
Writing x - 2 = X and y = Y, the given equation becomes
`"X"^2/12 - "Y"^2/4 = 1`
Here, a2 = 12 and b2 = 4 ⇒ a = `2sqrt3` and b = 2
For centre put X = 0 and Y = 0
x - 2 = 0 and y = 0
x = 2 and y = 0
∴ Coordinates of the centre are (2,0)
For foci, X = ± ae and Y = 0
Now, e = `sqrt("a"^2 + "b"^2)/"a"^2 = sqrt(12 + 4)/12 = sqrt(16/12) = 4/(2sqrt3) = 2/sqrt3`
∴ x - 2 = ± `2sqrt3 xx 2/sqrt3 = +-4`
x = ± 4 + 2
⇒ x = 6 , x = -2 and y = 0
∴ Coordinates of Foci are (6,0) and (-2,0)
The equation of directrices are:
X = `+- "a"/"e"`
x - 2 = `+- (2sqrt3)/(2/sqrt3)`
x - 2 = ± 3
x = ± 3 + 2 ⇒ x = 5 and x = -1
∴ x = 5 and x = -1 are the equations of directrices.
APPEARS IN
संबंधित प्रश्न
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
y (1 + ex) dy = (y + 1) ex dx
(y + xy) dx + (x − xy2) dy = 0
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Form the differential equation from the relation x2 + 4y2 = 4b2
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`dy/dx + 2xy = x`
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
