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
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 = c1e2x + c2e5x
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
Solve the following differential equation:
cos x . cos y dy − sin x . sin y dx = 0
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
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"`.
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
The differential equation having y = (cos-1 x)2 + P (sin-1 x) + Q as its general solution, where P and Q are arbitrary constants, is
Find the differential equation of the family of all non-horizontal lines in a plane
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
If `x^2 y^2 = sin^-1 sqrt(x^2 + y^2) + cos^-1 sqrt(x^2 + y^2)`, then `"dy"/"dx"` = ?
The elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant) ______.
The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is ______.
The differential equation of all circles passing through the origin and having their centres on the X-axis is ______.
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
Form the differential equation of all concentric circles having centre at the origin.
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.
