A) \[\frac{{{4}^{4}}}{5}\]
B) \[\frac{{{5}^{4}}}{4}\]
C) \[\frac{{{4}^{4}}}{{{5}^{4}}}\]
D) \[{{\left( \frac{5}{4} \right)}^{4}}\]
Correct Answer: C
Solution :
The given curve is\[4{{x}^{5}}=5{{y}^{4}}\] \[\Rightarrow \] \[20{{x}^{4}}=20{{y}^{3}}.\frac{dy}{dx}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{{{x}^{4}}}{{{y}^{3}}}\] We know that Length of subnormal \[(SN)=\left( y.\frac{dx}{dy} \right)=\left( \frac{{{y}^{4}}}{{{x}^{4}}} \right)\] Length of subtangent\[(ST)=\left( y.\frac{dy}{dx} \right)=\left( \frac{{{x}^{4}}}{{{y}^{2}}} \right)\] But given condition is \[\frac{{{(SN)}^{3}}}{{{(ST)}^{2}}}=\frac{{{({{y}^{4}}/{{x}^{4}})}^{3}}}{{{({{x}^{4}}/{{y}^{2}})}^{2}}}={{\left( \frac{{{y}^{4}}}{{{x}^{4}}} \right)}^{3}}\times {{\left( \frac{{{y}^{2}}}{{{x}^{4}}} \right)}^{2}}\] \[=\frac{{{y}^{12}}}{{{x}^{12}}}\times \frac{{{y}^{4}}}{{{x}^{8}}}=\left( \frac{{{y}^{16}}}{{{x}^{20}}} \right)\] \[={{\left( \frac{{{y}^{4}}}{{{x}^{5}}} \right)}^{4}}\] \[={{\left( \frac{4}{5} \right)}^{4}}=\frac{{{4}^{4}}}{{{5}^{4}}}\] \[(\because 4{{x}^{5}}=5{{y}^{4}})\]
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