Q.(a) For what value of $k$ does the equation $12x^2+2kxy+2y^2+11x-5y+2=0$ represent two straight lines. OR

Applying the standard "pair of straight lines" condition $abc+2fgh-af^2-bg^2-ch^2=0$ to the given equation gives the quadratic $4k^2+55k+175=0$, whose roots are $k=-5$ or $k=-\dfrac{35}{4}$.

Write the general second-degree equation as $ax^2+2hxy+by^2+2gx+2fy+c=0$. Comparing with $12x^2+2kxy+2y^2+11x-5y+2=0$:

$a=12,\ h=k,\ b=2,\ g=\dfrac{11}{2},\ f=-\dfrac52,\ c=2$.

The equation represents a pair of straight lines when $abc+2fgh-af^2-bg^2-ch^2=0$.

$abc = 12\times2\times2=48$.

$2fgh = 2\left(-\dfrac52\right)\left(\dfrac{11}{2}\right)k = -\dfrac{55}{2}k$.

$af^2 = 12\times\dfrac{25}{4}=75$.

$bg^2 = 2\times\dfrac{121}{4}=\dfrac{121}{2}=60.5$.

$ch^2 = 2k^2$.

Condition: $48-\dfrac{55}{2}k-75-60.5-2k^2=0 \Rightarrow -2k^2-\dfrac{55}{2}k-87.5=0$.

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