Question

If 3x²+xy-y²-3x+6y+k =0 , represents a pair of line , then k is equal to?
​

Answer 1

3x

2

+xy−y

2

−3x+6y+k=0

Condition for pair of lines:

Δ=

∣

∣

∣

∣

∣

∣

∣

∣

a

h

g

h

b

f

g

f

c

∣

∣

∣

∣

∣

∣

∣

∣

=0

comparing the equation with standard equation

ax

2

+2hxy+by

2

+2gx+2fy+c=0

⇒a=3;b=−1;c=k;h=

2

1

;g=

2

−3

;f=3

So, substituting the values in the above condition:

Δ=

∣

∣

∣

∣

∣

∣

∣

∣

3

1/2

−3/2

1/2

−1

3

−3/2

3

k

∣

∣

∣

∣

∣

∣

∣

∣

=0

⇒3(−k−9)−

2

1

(

2

k

+

2

9

)−

2

3

(

2

3

−

2

3

)=0

3(−k−9)−

4

1

(k+9)=0

⇒k=−9