Question

If the line $ax+by+c=0$ is normal to the $xy+5=0$, then a and b have

• A. same sign
• B. opposite sign
• C. Cannot be discussed
• D. None of these

Answer 1

The correct option is A same sign

Given,

$ax+by+c=0$

and $xy+5=0$

$x\frac{dy}{dx}+y=0$

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

${\frac{dy}{dx}}_{(x_1,y_1)}=-\frac{y_1}{x_1}$

$y-y_1=\left(\frac{x_1}{y_1}\right)(x-x_1)$

$y{y}_1-y_1^2=x{x}_1-x_1^2$

$ax+by+c=0$

$\to a=x_1,b=-y_1$

$x_1{y}_1+5=0$

$x_1{y}_1=-5<0$

$x_1{y}_1<0$

$x_1<0,y_1>0\Rightarrow a<0,b<0$

$x_1>0,y_1<0\Rightarrow a>0,b>0$