Correct Answer - Option 2 : 1
Concept:
Differentiate the x and y with respect to \(θ\)
\(\Rightarrow \frac{\mathrm{d}y }{\mathrm{d} x}=\frac{\frac{\mathrm{d} y}{\mathrm{d} θ }}{\frac{\mathrm{d} x}{\mathrm{d} θ }}\)
Calculation:
Given: \(x=a(θ +sinθ)\)
Differentiation with respect to θ
\(\Rightarrow \frac{\mathrm{d} x}{\mathrm{d} θ }=a(1+cosθ )\)
Now, \(y=a(3-cos\theta)\)
Differentiation with respect to θ
\(\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} θ }=a(sinθ )\)
\(\Rightarrow \frac{\mathrm{d}y }{\mathrm{d} x}=\frac{\frac{\mathrm{d} y}{\mathrm{d} θ }}{\frac{\mathrm{d} x}{\mathrm{d} θ }}\)
\(\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{sinθ }{1+cosθ }\)
Putting the value of \(θ=\pi/2\) we get
\(\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{sin\frac{\pi}{2}}{1+cos\frac{\pi}{2}}\)
\(\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}_{θ =\frac{\pi}{2}}=1\)
Hence, option 2 is correct.
