Advertisements
Advertisements
प्रश्न
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
Advertisements
उत्तर
Given circle equation be x2 + y2 = r2
Let y = mx + c be all straight lines which towards the given circle x2 + y2 = r2
The condition for y = mx + c ……. (1)
Be a tangent to the circle x2 + y2 = r2
Be c2 = r2(1 + m2)
⇒ c = `sqrt(1 + "m"^2)`
Substituting c value in equation (1), we get
y = `"mx" + "r" sqrt(1 + "m"^2)`
y – mx = `"r" sqrt(1 + "m"^2)` ......(2)
Differentiating equation (2) w.r.t x, we get
`("d"y)/("d"x) - "m"` = 0
`("d"y)/("d"x)` = m ........(3)
Substituting equation (3) in equation (2), we get
`y - x(("d"y)/("d"x)) = "r" sqrt(1 + (("d"y)/("d"x))^2`
Squaring on both sides, we get
`[y - x ("d"y)/"d"x]^2 = ["r" sqrt(1 + (("d"y)/("d"x))^2]]^2`
`[y - x ("d"y)/"d"x]^2 = "r"^2 [1 + (("d"y)/("d"x))^2]`
Which is a required differential equation.
APPEARS IN
संबंधित प्रश्न
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
For the following differential equation find the particular solution satisfying the given condition:
`y(1 + log x) dx/dy - x log x = 0, y = e^2,` when x = e
For the following differential equation find the particular solution satisfying the given condition:
`(e^y + 1) cos x + e^y sin x. dy/dx = 0, "when" x = pi/6,` y = 0
Reduce the following differential equation to the variable separable form and hence solve:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
Reduce the following differential equation to the variable separable form and hence solve:
(2x - 2y + 3)dx - (x - y + 1)dy = 0, when x = 0, y = 1.
In the following example verify that the given function is a solution of the differential equation.
`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Find the particular solution of the following differential equation:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
Select and write the correct alternative from the given option for the question
The solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
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 from the relation x2 + 4y2 = 4b2
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
Choose the correct alternative:
The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
