Exercise 11.5
Question 1
\frac{d y}{d x} ज्ञात करे जब[Find \frac{d y}{d x} when]
(i) x=acosθ , y=asinθ
Sol :
Differentiate w.r.t θ
\frac{d x}{d \theta}=-a \sin \theta..(i)
\frac{d y}{d \theta}=a \cos \theta..(ii)
समीकरण (i) मे (ii) से भाग देने पर ,
\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{a \cos \theta}{- a \sin \theta}
=-cotθ
(ii) x=acosθ , y=bcosθ
Sol :
Differentiate w.r.t θ
\frac{d x}{d \theta}=-a \sin \theta..(i)
\frac{d y}{d \theta}=-b \sin \theta..(ii)
समीकरण (i) मे (ii) से भाग देने पर ,
\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{-b \sin \theta}{- a \sin \theta}
\frac{d y}{d x}=\frac{b}{a}
Question 2
\frac{d y}{d x} ज्ञात करे जब[Find \frac{d y}{d x} when]
Sol :
Differentiate w.r.t t
\frac{d x}{d t}=2 a t..(i)
\frac{d y}{d t}=2 a..(ii)
समीकरण (ii) मे (i) से भाग देने पर ,
\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{2a}{2at}
\frac{d y}{d x}=\frac{1}{t}
(ii) x=4t , y=\frac{4}{t}
Sol :
Differentiate w.r.t t
\frac{d x}{d t}=4..(i)
\frac{d y}{d t}=-\frac{4}{t^{2}}..(ii)
समीकरण (ii) मे (i) से भाग देने पर ,
\dfrac{\frac{d y}{d t}}{\frac{dx}{dt}}=\dfrac{\frac{-4}{t^2}}{4}
\frac{d y}{d x}=\frac{-1}{t^{2}}
Question 3
\frac{d y}{d x} ज्ञात करे जब[Find \frac{d y}{d x} when]
Sol :
Differentiate w.r.t t
\frac{d x}{d t}=\cos \theta..(i)
\frac{d y}{d t}=-\sin 2 t \times 2
\frac{dy}{d t}=-2 .\sin 2t
समीकरण (ii) मे (i) से भाग देने पर ,
\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{-2 \cdot \cos t}{\cos t}
\frac{d y}{d x}=\frac{-2 \times 2\sin t \cos t}{\cos t}
⇒-4sint
(ii) x=asecθ,y=btanθ
Sol :
Differentiating w.r.t θ
\frac{d x}{d \theta}=a \sec \theta \tan \theta..(i)
\frac{d y}{d \theta}=b \sec ^{2} \theta..(ii)
समीकरण (ii) मे (i) से भाग देने पर ,
\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{b \sec ^{2} \theta}{a \sec \theta \tan \theta}
\frac{d y}{d x}=\frac{b}{a}\times \frac{1}{\cos \theta}
=\frac{b}{a} \operatorname{cosec} \theta
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