Question

The slope of normal at any point $P$ of a curve (lying in the first quadrant) is reciprocal of twice the product of the abscissa and the ordinate of point $P.$ Then, the equation of the curve is (where, $c$ is an arbitrary constant)

• A. $y^{2}=x+c$
• B. $y = c e^{- x^{2}}$
• C. $y=ce^{- x}$
• D. $y^{2}=ln x+c$

Answer 1

Answer: (B)

Slope of normal at $P=\frac{- 1}{\left(\frac{d y}{d x}\right)_{P}}$
$\Rightarrow \frac{1}{2 x y}=\frac{- 1}{\left(\frac{d y}{d x}\right)}$
or $\frac{d y}{d x}=-2xy$
$\Rightarrow \displaystyle \int \frac{d y}{y}=\displaystyle \int - 2 xdx$
$\Rightarrow ln y=-x^{2}+k$
$\Rightarrow y = e^{- x^{2} + k} = c e^{- x^{2}}$