Question

The equation of the curve passing through the point (1, 1) such that the slope of the tangent at any point (x, y) is equal to the product of its co-ordinates is

• A. $2 \log y = x^2 + 1$
• B. $ 2 \log x = y^2 - 1$
• C. $2 \log x =y^2 + 1 $
• D. $ 2 \log y = x^2 - 1 $

Answer 1

Answer: (D)

$\frac{d y}{d x}=xy$
$\frac{1}{y} dy=xdx$
$Integrate$
$log y=\frac{x^{2}}{2}+c\quad\ldots\left(1\right)$
$Equation \left(1\right) passing \,through \left(1, 1\right)$
$log\left(1\right)=\frac{1^{2}}{2}+C$
$0=\frac{1}{2}+c$
$c=-\frac{1}{2}$
$log\, y=\frac{x^{2}}{2}-\frac{1}{2}$
$2 \,log \, y=x^{2}-1$