Question

Find the condition that the slope of one of the lines represented by $a{x}^2+2hxy+b{y}^2=0$ should be $n$ times the slope of the other.

Answer 1

Let $l_{1}x+m_{1}y=0$ $\&$ $l_{2}x+m_{2}y=0$ be the lines represented by $a{x}^2+2hxy+b{y}^2=0$

$\Rightarrow (l_{1}x+m_{1}y)(l_{2}x+m_{2}y)=a{x}^2+2hxy+b{y}^2$

$\Rightarrow l_1{l}_2=a;$ $m_1{m}_2=b;$ $l_1{m}_2+l_2{m}_1=2h$

Now, slope of $l_{1}x+m_{1}y=0=\frac{-l_1}{m_1}$

$\&$ slope of $l_{2}x+m_{2}y=\frac{-l_2}{m_2}$

According to questions -

$\Rightarrow \frac{-l_1}{m_1}=n.\frac{-l_2}{m_2}$

$l_1{m}_2=n.m_1{l}_2$

$l_1{m}_2+l_2{m}_1=(1+n)l_1{m}_2$

$2h=(1+n)l_1{m}_2$

Also, $l_1{l}_2=m_1{m}_2=ab$

$\Rightarrow l_1{m}_2.n.l_1{m}_2=ab$

$\Rightarrow n{\left(l_1{m}_2\right)}^2=ab$

$\Rightarrow l_1{m}_2=\sqrt{\frac{ab}{n}}$

$\Rightarrow 2h=(1+n)\sqrt{\frac{ab}{n}}$ required condition.

Hence, solved.