Question
Download Solution PDFयदि x = k(θ + sin θ) और y = k (1 + cos θ) है, तो θ = π/2 पर x के संबंध में y का अवकलज क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
यदि x = f(θ), y = f(θ) है, तो,
श्रृंखला नियम से, हमारे पास निम्न है
\(\rm \frac {dy}{dx} = \dfrac {\frac {dy}{d\theta}}{\frac {dx}{d\theta}}\)
गणना:
दिया गया है, x = k(θ + sin θ)
⇒\(\rm \dfrac {dx}{dθ} = k (1+cos \;θ)\)
y = k (1 + cos θ)
⇒\(\rm \dfrac {dy}{dθ} = k (-sin \;θ)\)
श्रृंखला नियम से, हमारे पास निम्न है
\(\rm \dfrac {dy}{dx} = \dfrac {\dfrac {dy}{d\theta}}{\dfrac {dx}{d\theta}}\)
⇒ \(\rm \dfrac{dy}{dx} = \dfrac {-ksin \;\theta}{k(1+ cos\;\theta)}\)
⇒ \(\rm \dfrac{dy}{dx} = \dfrac {-sin \;\theta}{(1+ cos\;\theta)}\)
θ = π/2 रखने पर
\(\rm \dfrac{dy}{dx} = \dfrac {-sin \;\dfrac {\pi}{2}}{(1+ cos\;\dfrac {\pi}{2})}\)
⇒ \(\rm \dfrac{dy}{dx} = -1\)
अतः यदि x = k(θ + sin θ) और y = k (1 + cos θ) है, तो θ = π/2 पर x के संबंध में y का अवकलज -1 है।
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