If $x=a\sin 2t(1+\cos 2t)$ and $y=b\cos 2t(1-\cos 2t)$, show that $\left(\dfrac{dy}{dx}\right)_{t=\pi/4}=\dfrac{b}{a}$.

CBSE Class 12 · DifferentiationLong Answer

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

and

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

, show that

(\frac{d y}{d x})_{t = π /4} = \frac{b}{a}

Answer: $\frac{b}{a}$ (proved)

Expand and use double-angle forms. For

x

x = a sin 2 t + \frac{a}{2} sin 4 t \Rightarrow \frac{d x}{d t} = 2 a cos 2 t + 2 a cos 4 t

For

y

(with

cos^{2} 2 t = \frac{1 + c o s 4 t}{2}

y = b cos 2 t - \frac{b}{2} - \frac{b}{2} cos 4 t \Rightarrow \frac{d y}{d t} = - 2 b sin 2 t + 2 b sin 4 t

Hence:

\frac{d y}{d x} = \frac{2 b ( sin 4 t - sin 2 t )}{2 a ( cos 4 t + cos 2 t )} = \frac{b ( sin 4 t - sin 2 t )}{a ( cos 4 t + cos 2 t )}

t = \frac{π}{4}

sin 4 t = sin π = 0, sin 2 t = sin \frac{π}{2} = 1, cos 4 t = cos π = - 1, cos 2 t = cos \frac{π}{2} = 0

\frac{d y}{d x} = \frac{b ( 0 - 1 )}{a ( - 1 + 0 )} = \frac{b}{a}

(\frac{d y}{d x})_{t = π /4} = \frac{b}{a}

CBSE Class 12 Board · Subjective