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Question

Which among the following equations represents a pair of straight lines?


A

x2 - y2 + x - y + 1 = 0

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B

x2 - 2x - y + 3 = 0

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C

x2 + 3xy + 2y2 - x - 4y - 6 = 0

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D

3x2 - 4xy - 7y2 = 0

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Solution

The correct option is D

3x2 - 4xy - 7y2 = 0


Equations of the form ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 can be the equation of a conic (Circle, Parabola, Ellipse or Hyperbola) or pair of straight lines.

To decide if it represents a conic or pair of straight lines, we will find

= abc + 2fgh af2 bg2 ch2 = 0

If = 0, the given equation will represent a pair of straight lines. We will find for each equation and decide.

1) x2 y2 + x y + 1 = 0

a = 1,b = 1,h = 0,g = 12,f = 12,c = 1

= 1 × 1 × 1 + 0 (12)2 + (12)2 1 × 0 = 1

0 conic

2) x2 2x y + 3 = 0

a=1,b=0,h=0,g=1,f=12,c=3

= 0 + 0 1 × (12)2 0 0

=14

0 conic

3) x2 + 3xy + 2y2 x 4y 6 = 0

a = 1,b=2,h=32,g=12,f=2,c=6

= 1 × 2 × 32 + 2 × 2 × 12 × 32 1 × (2)2 2 × (12)2 (6) × (32)2

3+3412 + 272

=94+13

0 conic

4) 3x2 4xy 7y2 = 0

a=3,b=7,h=2,g=0,f=c=0

= 0 pair of straight lines

So, only the last equation represents a pair of straight lines.


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