A) \[{{\text{(Subnormal)}}^{1/2}}\]
B) Subnormal
C) \[{{\text{(Subnormal)}}^{\text{3/2}}}\]
D) None of these
Correct Answer: B
Solution :
\[b{{y}^{2}}={{(x+a)}^{3}}\Rightarrow 2by.\frac{dy}{dx}=3{{(x+a)}^{2}}\Rightarrow \frac{dy}{dx}=\frac{3}{2by}{{(x+a)}^{2}}\] \ Subnormal = \[y\frac{dy}{dx}=\frac{3}{2b}{{(x+a)}^{2}}\] \ Subtangent = \[\frac{y}{\left( \frac{dy}{dx} \right)}=\frac{y}{\frac{3{{(x+a)}^{2}}}{2by}}=\frac{2b{{y}^{2}}}{3{{(x+a)}^{2}}}\] = \[\frac{2b{{\frac{(x+a)}{b}}^{3}}}{3{{(x+a)}^{2}}}=\frac{2}{3}(x+a)\] \ (Subtangent)2 = \[\frac{4}{9}{{(x+a)}^{2}}\] and \[\frac{{{(\text{Subtangent})}^{2}}}{\text{Subnormal}}=\frac{\frac{4}{9}{{(x+a)}^{2}}}{\frac{3}{2b}{{(x+a)}^{2}}}=\frac{8b}{27}\] Þ (Subtangent)2 = constant ´ (Subnormal). \ (Subtangent)2 µ (Subnormal).
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