Applications Of Derivatives Question 156
Question: . If ST and SN are the lengths of the subtangent and the subnormal at the point $ \theta =\frac{\pi }{2} $ on the curve $ x=a(\theta +\sin \theta ),y=a(1-\cos \theta ),a\ne 1 $ , then
[Karnataka CET 2005]
Options:
A) $ ST=SN $
B) $ ST=2,SN $
C) $ ST^{2}=a,SN^{3} $
D) $ ST^{3}=a,SN $
Show Answer
Answer:
Correct Answer: A
Solution:
$ \frac{dx}{d\theta }=a(1+\cos \theta ),,\frac{dy}{d\theta }=a,(\sin \theta ) $
$ {{. \frac{dy}{dx} |} _{\theta =\frac{\pi }{2}}}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=\frac{a\sin \theta }{a(1+\cos \theta )}=1, $
$ {{. y |} _{\theta =\frac{\pi }{2}}}=a $
Length of sub-tangent ST = $ \frac{y}{dy/dx}=\frac{a}{1}=a. $ and length of sub-normal SN = $ y\frac{dy}{dx}=a,.,1=a $
Hence $ ST=SN $ .
BETA
