Question

Find condition that the line $4x+5y=0$ coincides with one of the lines given by $a{x}^2+2hxy+b{y}^2=0$

Answer 1

The given equation is $a{x}^2+2hxy+b{y}^2=0$

We will first convert it into the auxiliary equation as follows:

Divide both sides by $x^2$, we get:

$a+2h\left(\frac{y}{x}\right)+b{\left(\frac{y}{x}\right)}^2=0......\left(1\right)$

Now, the given line equation is $4x+5y=0$

Thus $\frac{y}{x}=\frac{-4}{5}$

Now putting the above value in equation$\left(1\right)$, we will get:

$a+2h\left(\frac{-4}{5}\right)+b{\left(\frac{-4}{5}\right)}^2=0$

On simplifying,

$25a-40h+16b=0$ is the required condition.