Question

If $a{x}^2+2hxy+b{y}^2=0$ then $\frac{dy}{dx}$ is equal to

• A. $\frac{y}{x}$
• B. $\frac{x}{y}$
• C. $-\frac{x}{y}$
• D. none of these

Answer 1

The correct option is A $\frac{y}{x}$

$a{x}^2+2hxy+b{y}^2=0$

Put

$y=vx$

$\frac{dy}{dx}=x+x\frac{dv}{dx}$

$a{x}^2+2hx\left(vx\right)+b\left(v^2{x}^2\right)=0$

$\Rightarrow x^2[a+2hv+b{v}^2]=0$

$\Rightarrow b{v}^2+2hv+a=0$

$(2bv+2h)\frac{dv}{dx}=0$

$\Rightarrow \frac{dv}{dx}=0$

$\therefore \frac{dy}{dx}=v+x\left(0\right)=v$

$\therefore \frac{dy}{dx}=\frac{y}{x}$