Applications Of Derivatives Question 77
Question: If the relation between sub-normal SN and sub- tangent ST at any point S on the curve; $ by^{2}={{(x+a)}^{3}} $ is $ p(SN)=q{{(ST)}^{2}}, $ then the value of p/q is
Options:
A) 8a/27
B) 27/8b
C) 8b/27
D) 8/27
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Here, $ by^{2}={{(x+a)}^{3}} $ Differentiating both the sides, we get $ 2by\frac{dy}{dx}=3{{(x+a)}^{2}}\Rightarrow \frac{dy}{dx}=\frac{3{{(x+a)}^{2}}}{2by} $
$ \therefore $ length of subnormal $ SN=y\frac{dy}{dx}=\frac{3}{2}\frac{{{(x+a)}^{2}}}{b}\therefore $ length of subtangent $ ST=y.\frac{dx}{dy}=\frac{2by^{2}}{3{{(x+a)}^{2}}}\therefore p(SN)=q{{(ST)}^{2}} $
$ \Rightarrow \frac{p}{q}=\frac{{{(ST)}^{2}}}{(SN)}=\frac{8}{27}\frac{b^{3}y^{4}}{{{(x+a)}^{6}}}=\frac{8b}{27} $
$ ( \because \frac{b^{2}y^{4}}{{{(x+a)}^{6}}}=1 ) $
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