Question

Given that the slope of the tangent to a curve $y = y(x)$ at any point $(x,y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2 + y^2 - 2x - 2y = 0$, then its equation is :

• A. $x \log_e |y| = 2 (x - 1)$
• B. $ x \log_e |y| = x - 1$
• C. $x^2 \log_e |y| = - (x - 1)$
• D. $x \log_e |y| = - 2(x - 1)$

Answer 1

Answer: (A)

given $\frac{dy}{dx} = \frac{2y}{x^{2}} $
$ \Rightarrow \int \frac{dy}{2y} = \int \frac{dx}{x^{2}} $
$\Rightarrow \frac{1}{2} \ell ny =- \frac{1}{x} + c$
passes through centre (1,1)
$ \Rightarrow c = 1 $
$ \Rightarrow \; \; x \ell ny = 2\left(x-1\right) $