If x=a 2 t(1+ 2 t), y=b 2 t(1- 2 t) then d y/d x is equal to

Correct Answer is : batan⁡tb/a tabtant

Correct Answer is : batan⁡tb/a tabtant

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MHTCET 23 Apr 2025 Shift 1 Paper

Question : 136 of 150

Marks: +1, -0

x=a \sin 2 t(1+\cos 2 t), y=b \cos 2 t(1-\cos 2 t)

\frac{d y}{d x}

is equal to

Solution:

Concept:

Use parametric differentiation:

\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}

Explanation:

Given

x = a\sin 2t(1+\cos 2t)

and

y = b\cos 2t(1-\cos 2t)

Differentiate

x

with respect to

t

\dfrac{dx}{dt} = a\sin 2t(-2\sin 2t) + (1+\cos 2t)(2a\cos 2t)

= -2a\sin^2 2t + 2a\cos 2t + 2a\cos^2 2t

= 2a\cos 2t + 2a(\cos^2 2t - \sin^2 2t)

= 2a\cos 2t + 2a\cos 4t = 2a(\cos 2t + \cos 4t)

Differentiate

y

with respect to

t

\dfrac{dy}{dt} = b\cos 2t(2\sin 2t) + (1-\cos 2t)(-2b\sin 2t)

= 2b\sin 2t\cos 2t - 2b\sin 2t + 2b\sin 2t\cos 2t

= b\sin 4t - 2b\sin 2t + b\sin 4t

= 2b(\sin 4t - \sin 2t)

Now divide:

\dfrac{dy}{dx} = \dfrac{2b(\sin 4t - \sin 2t)}{2a(\cos 2t + \cos 4t)}

= \dfrac{b}{a} \cdot \dfrac{2\cos 3t \sin t}{2\cos 3t \cos t}

= \dfrac{b}{a}\tan t

Answer:

\dfrac{dy}{dx} = \dfrac{b}{a}\tan t

, which matches option A.

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