A) \[27\,{{c}^{3}}\]
B) \[\frac{4}{27}{{c}^{3}}\]
C) \[\frac{27}{4}{{c}^{3}}\]
D) \[\frac{4}{9}{{c}^{3}}\]
Correct Answer: C
Solution :
Given curve is\[{{x}^{2}}y={{c}^{3}}\]. Differentiating w.r.t. \[x,\]we get \[{{x}^{2}}\frac{dy}{dx}+2xy=0\] \[\Rightarrow \] \[\frac{dy}{dx}=-\frac{2y}{x}=-\frac{2y}{x}\] So, equation of tangent at\[(x,y)\]is: \[Y-y=-\frac{2y}{x}(X-x)\] \[Y=0,\]gives, \[X=\frac{3x}{2}=a\] and \[X=0,\]gives, \[Y=3y=b\] Now, \[{{a}^{2}}b=\frac{9{{x}^{2}}}{4}\cdot 3y=\frac{27}{4}{{x}^{2}}y=\frac{27}{4}{{c}^{3}}\]
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