A) \[8a/27\]
B) \[27/8b\]
C) \[8b/27\]
D) \[8/27\]
Correct Answer: C
Solution :
Here given curve, \[b{{y}^{2}}={{(x+a)}^{3}}\] |
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 is, \[SN=y\frac{dy}{dx}=\frac{3}{2}\frac{{{(x+a)}^{2}}}{b}\] and length of subtangent is, \[ST=y.\frac{dx}{dy}=\frac{2b{{y}^{2}}}{3{{(x+a)}^{2}}}\] |
\[\therefore \,\,p(SN)=q{{(ST)}^{2}}\] |
\[\Rightarrow \,\,\frac{p}{q}=\frac{{{(ST)}^{2}}}{(SN)}=\frac{8}{27}\frac{{{b}^{3}}{{y}^{4}}}{{{(x+a)}^{6}}}=\frac{8b}{27}\,\,\left( \because \,\,\frac{{{b}^{2}}{{y}^{4}}}{{{(x+a)}^{6}}}=1 \right)\] |
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