Question

For what value of k does the equation $12x^2-10xy+2y^2+11x-5y+k=0$ represents a pair of straight lines?

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Answer 1

Given equation is $12x^2-10xy+2y^2+11x-5y+k=0$

Comparing it with standard pair of straight lines

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

We get,

a=12, h=-5, b=2

$g=\frac{11}{2}$, $f=\frac{-5}{2}$, c=k

If equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a pair of straight line

Then $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

i.e., $\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$

Substituting values of a,b,c,h,g,f

We get,

$12\times 2\times k+2\left(\frac{-5}{2}\right)\left(\frac{11}{2}\right)(-5)-12(\frac{-5}{2}{)}^2-2(\frac{11}{2}{)}^2-k(-5{)}^2=0$

$24k+\frac{275}{2}-75-\frac{121}{2}-25k=0$

k - 2 = 0

k = 2