Question

Find the 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

Given joint equation of lines is

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

$\therefore$ It's auxilary equation is

$b{m}^2+2hm+a=0$ ______ (i)

One of the line is $4x+5y=0$

$\therefore$ slope = $\frac{-4}{5}=m$

Now, $-4/5$ must be one of the roots of the auxilary equation (i)

$\therefore b(-4/5{)}^2+2b(-4/5)+a=0$

$b\left(16/25\right)+2h(-4/5)+a=0$

multiplying both sides of equation by 25

$\therefore 16b-40h+25a=0$

$\therefore 25a-40b+16=0$

Hence proved