question_answer

If area of triangle formed by tangent (with positive slope) and normal to a curve at any point in first quadrant with x-axis is cube of its ordinate, then differential equation of such a family of curves is

A) \[{{\left( \frac{dy}{dx} \right)}^{2}}=2y\frac{dy}{dx}\]

B)        \[{{\left( \frac{dy}{dx} \right)}^{2}}-2y\frac{dy}{dx}+1=0\]

C) \[2y{{\left( \frac{dy}{dx} \right)}^{2}}+1=0\]

D) \[{{\left( \frac{dy}{dx} \right)}^{2}}+2y\frac{dy}{dx}+1=0\]

Correct Answer: B

Solution :

[b] Area \[=\frac{1}{2}(ST+SN)\times PS\] \[\Rightarrow \,\,\,\,{{y}^{3}}=\frac{1}{2}\left( yy'+\frac{y}{y'} \right)y\]   (Given) \[\Rightarrow \,\,\,\,2y=y'+\frac{1}{y'}\] \[\Rightarrow \,\,\,\,{{\left( \frac{dy}{dx} \right)}^{2}}-2y\left( \frac{dy}{dx} \right)+1=0\]