the condition for a conic x2+2xy+2y+kx+3y2=0 to represent a pair of straight lines
This is a straight forward condition. For a conic of the general form,
ax2+2hxy+2gx+2fy+by2=0 to represent a pair of straight line, the required condition in determinant form is,
∣∣ ∣∣ahghbfgfc∣∣ ∣∣ =0
The given conic equation is,
x2+2xy+2y+kx+3y2=0
Where,
a =1 , b = 1 , h = 1 , g = k2 , f = 1, c = 0;
Putting these values in the condition we get,
∣∣ ∣ ∣∣11k2111k211∣∣ ∣ ∣∣ =0
∣∣ ∣ ∣∣11k2−1111k211∣∣ ∣ ∣∣ =0
(k2−1)(1−k2)=0
k2=1,
k= 2, which is the required condition.