Find k, if the slopes of the lines given by $k x^{2} + 5 x y + y^{2} = 0$ differ by 1.

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Solution

We know that ,
for $a x^{2} + 2 h x y + b y^{2} = 0$

$m_{1} + m_{2} = \frac{- 2 h}{b} m_{1} m_{2} = \frac{a}{b}$

So here, $m_{1} + m_{2} = - 5 m_{1} m_{2} = k$

and it is given that, $m_{1} - m_{2} = 1$

So, $(m_{1} - m_{2})^{2} = (m_{1} + m_{2})^{2} - 4 m_{1} m_{2} \Rightarrow (1)^{2} = (- 5)^{2} - 4 k \Rightarrow 4 k = 24 \Rightarrow k = 6$

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