Question

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

Solution

## The correct option is B k=2This 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.

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