# For what value of k will the equation 12x^2-10xy+2y^2+11x-5y+k=0 represent a pair of lines?

Aug 8, 2018

$k = 2$

#### Explanation:

We know that ,
color(blue)("If " ax^2+2hxy+by^2+2gx+2fy+c=0, a^2+h^2+b^2!=0

color(blue)("represent a pair of lines",then

color(blue)(D=|(a,h,g),(h,b,f),(g,f,c)|=0

We have,

$12 {x}^{2} - 10 x y + 2 {y}^{2} + 11 x - 5 y + k = 0$

$\therefore a = 12 , h = - 5 , b = 2 , g = \frac{11}{2} , f = - \frac{5}{2} \mathmr{and} c = k$

$\therefore D = | \left(12 , - 5 , \frac{11}{2}\right) , \left(- 5 , 2 , - \frac{5}{2}\right) , \left(\frac{11}{2} , - \frac{5}{2} , k\right) | = 0$

Expanding we get

$12 \left(2 k - \frac{25}{4}\right) + 5 \left(- 5 k + \frac{55}{4}\right) + \frac{11}{2} \left(\frac{25}{2} - 11\right) = 0$

$24 k - 75 - 25 k + \frac{275}{4} + \frac{275}{4} - \frac{121}{2} = 0$

$\therefore - k - 75 + \frac{275 + 275 - 242}{4} = 0$

$\therefore - k - 75 + 77 = 0$

$\therefore - k + 2 = 0$

$\therefore k = 2$