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Question

the condition for a conic x2+2xy+2y+kx+3y2=0 to represent a pair of straight lines 
  1. k=1
  2. k=2
  3. k=3
  4. k=4


Solution

The correct option is B k=2

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

∣ ∣ ∣11k21111k211∣ ∣ ∣ =0

(k21)(1k2)=0

k2=1,

k= 2, which is the required condition.

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