Find the value of k so that the equation of pair of line x²+kxy+2y²+3x+5y+2=0

the condition for a general quadratic equation of the form ax^2+by^2+2gx+2fy+2hxy+c=0 to represent a pair of straight lines (POSL) is that the determinant should be 0.

Here,

a=1, b= 2, g= 3/2, f= 5/2, h= k/2, c=2

so we form the determinant as follows:

|  1       k/2    3/2|

| k/2      2      5/2| = 0

| 3/2    5/2       2 |

putting this determinant 0, we get

 K= 3, 9/2