Question

Find k, if the slopes of the lines represented by 3x^2 + kxy - y^2 = 0 differ by 4.

Answer 1

Answer:

$\textbf{The value of k is {\pm}2 }$

Step-by-step explanation:

$\textbf{Given:}$

$3x^2+kxy-y^2=0$

$\text{Here,}a=3,\;2h=k,\;b=-1$

$\text{Let m_1 and m_2 be the slopes of the given lines}$

$\text{Then,}$

$m_1+m_2=\frac{-2h}{b}\;\;\&\;\;m_1\,m_2=\frac{a}{b}$

$\implies\,m_1+m_2=\frac{-k}{-1}\;\;\&\;\;m_1\,m_2=\frac{3}{-1}$

$\implies\,m_1+m_2=k\;\;\&\;\;m_1\,m_2=-3$

$\text{using,}$

$\bf\,(a-b)^2=(a+b)^2-4ab$

$\impies(m_1-m_2)^2=(m_1+m_2)^2-4\,m_1\,m_2$

$\impies\,4^2=k^2-4(-3)$

$\impies\,16=k^2+12$

$\impies\,k^2=4$

$\implies\boxed{\bf\,k={\pm}2}$