Advertisements
Advertisements
प्रश्न
If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1−cos 2t), show that `dy/dx=β/αtan t`
Advertisements
उत्तर
`x=α sin 2t(1+cos 2t)`
`⇒x=α sin 2t+α/2×2 sin 2tcos 2t`
`⇒x=α sin 2t+α/2 sin 4t`
Differentiating both sides w.r.t. t, we get
`dx/dt=α cos2t xx 2+α/2 cos4txx4`
`⇒dx/dt=2α(cos2t+cos4t)`
`⇒dx/dt=2α(cos2t+2cos^2 2t−1)`
`⇒dx/dt=2α(cos2t+1)(2cos2t−1)`
Now,
`y=β cos2t(1−cos2t)`
`⇒y=β cos2t−β cos^2 2t`
Differentiating both sides w.r.t. t, we get
`dy/dt=−β sin2t xx 2+β xx 2cos2t xx sin2txx2`
`⇒dy/dt=−2β sin2t+4β cos2t.sin2t`
`⇒dy/dt=2β sin2t(2cos2t−1)`
We know
`dy/dx=(dy/dt)/(dx/dt)`
`=(2α(cos2t+1)(2cos2t−1))/(dy/dt=2β sin2t(2cos2t−1))`
`=(βsin2t)/(α(cos2t+1))`
`=(βxx2sintcost)/(αxx2cos^2t)`
`=β/αtant`
APPEARS IN
संबंधित प्रश्न
If `log_10((x^3-y^3)/(x^3+y^3))=2 "then show that" dy/dx = [-99x^2]/[101y^2]`
If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and hence, find dy/dx if x=a cost, y=a sint
If x=at2, y= 2at , then find dy/dx.
If y =1 − cos θ, x = 1 − sin θ, then `dy/dx "at" θ =pi/4` is ______
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 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `at t = `pi/4`
Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = 2at2, y = at4
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find `dy/dx` when `theta = pi/3`
If X = f(t) and Y = g(t) Are Differentiable Functions of t , then prove that y is a differentiable function of x and
`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`
Hence find `"dy"/"dx"` if x = a cos2 t and y = a sin2 t.
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)`
x = `"t" + 1/"t"`, y = `"t" - 1/"t"`
x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`
x = 3cosθ – 2cos3θ, y = 3sinθ – 2sin3θ
sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`
Derivative of x2 w.r.t. x3 is ______.
If y `= "Ae"^(5"x") + "Be"^(-5"x") "x" "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
When \(x\) and \(y\) are expressed separately as functions of the same third variable \(t\), which equations are obtained?
In the parametric equations \(x=f(t)\) and \(y=g(t)\), what is \(t\) called?
Under what condition is the formula \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\) directly applicable?
After writing \(x=f(t)\) and \(y=g(t)\), what should be found separately?
For \(x=a\cos^3\theta\) and \(y=a\sin^3\theta\), which equation do these parametric equations represent?
If \(x=a\cos t\) and \(y=a\sin t\), what is \(\frac{dy}{dx}\)?
