Advertisements
Advertisements
Question
If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`
Solution
`x=a(t-1/t),y=a(t+1/t)`
`x/a=t-1/t and y/a=t+1/t`
we have
`(t+1/t)^2=(t-1/t)^2+4`
`(y/a)^2=(x/a)^2+4`
`y^2/a^2-x^2/a^2=4`
`y^2-x^2=4a^2`
Differentiating w.r.t. x
`2y dy/dx-2x=0`
`dy/dx=2x/2y`
`dy/dx=x/y`
APPEARS IN
RELATED QUESTIONS
find dy/dx if x=e2t , y=`e^sqrtt`
If x=at2, y= 2at , then find dy/dx.
If `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`
If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `
If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find `dy/dx `
If x = cos t (3 – 2 cos2 t) and y = sin t (3 – 2 sin2 t), find the value of dx/dy at t =4/π.
Derivatives of tan3θ with respect to sec3θ at θ=π/3 is
(A)` 3/2`
(B) `sqrt3/2`
(C) `1/2`
(D) `-sqrt3/2`
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`
`x = 2at^2, y = at^4`
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx.`
x = sin t, y = cos 2t
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx.`
`x = 4t, y = 4/y`
If x and y are connected parametrically by the equation without eliminating the parameter, find `dy/dx`.
x = cos θ – cos 2θ, y = sin θ – sin 2θ
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx.`
`x = (sin^3t)/sqrt(cos 2t), y = (cos^3t)/sqrt(cos 2t)`
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx.`
`x = a(cos t + log tan t/2), y = a sin t`
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx.`
x = a sec θ, y = b tan θ
IF `y = e^(sin-1x) and z =e^(-cos-1x),` prove that `dy/dz = e^x//2`
The cost C of producing x articles is given as C = x3-16x2 + 47x. For what values of x, with the average cost is decreasing'?
If y = sin -1 `((8x)/(1 + 16x^2))`, find `(dy)/(dx)`
Evaluate : `int (sec^2 x)/(tan^2 x + 4)` dx
x = `"t" + 1/"t"`, y = `"t" - 1/"t"`
sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`
If x = ecos2t and y = esin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`
If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at t" = pi/4) = "b"/"a"`
If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`
Differentiate `x/sinx` w.r.t. sin x
If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0
Derivative of x2 w.r.t. x3 is ______.
If `"x = a sin" theta "and y = b cos" theta, "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
Form the point of intersection (P) of lines given by x2 – y2 – 2x + 2y = 0, points A, B, C, Dare taken on the lines at a distance of `2sqrt(2)` units to form a quadrilateral whose area is A1 and the area of the quadrilateral formed by joining the circumcentres of ΔPAB, ΔPBC, ΔPCD, ΔPDA is A2, then `A_1/A_2` equals
Let a function y = f(x) is defined by x = eθsinθ and y = θesinθ, where θ is a real parameter, then value of `lim_(θ→0)`f'(x) is ______.
