Question

The slope of the tangent at $(x,y)$ of a curve passing through the point $(2,2)$ is $\frac{x}{y}$,then the equation of the curve is

• A. $x^2+y^2=4$
• B. $x^2-y^2=4$
• C. $x^2=y^2$
• D. $x^2=y^2-4$

Answer 1

Answer: (C)

$x^2=y^2$

Given, $\frac{dy}{dx}=\frac{x}{y}$

$\Rightarrow ydy=xdx$

$\Rightarrow {\int }^{}ydy={\int }^{}xdx$

$\Rightarrow \frac{y^2}{2}=\frac{x^2}{2}+c$

$\Rightarrow y^2=x^2+2c$ Since this curve passes through point $(2,2)$

Therefore, ${\left(2\right)}^2={\left(2\right)}^2+2c$

$\Rightarrow c=0$

putting value of $c$ in above equation of curve we get

$y^2=x^2$

Hence, The required equation of the curve is $x^2=y^2$