Question

Write conditions for line represented by equation a1x+b1y+c1=0and a2x + b2y+c2=0 to be parallel

Answer 1

Step-by-step explanation:

• When two lines cross in the same direction, they are either vertical or non-vertical.
• If they're both vertical, b1 = b2 = 0, then a1b2 = a2b1 = 0.
• Their slopes are same if both are non-vertical.

Thus,

- a1/b1 = -a2/b2

Hence,

a1b2 = a2b1 or a1:b1 = a2:b2

• When a1b2 = a2b1 and b1 = 0, a1 = 0 and b1 = 0, a1 = 0 and b2 = 0.
• As a result, b1 = 0 b2 = 0, and if one of the lines is vertical, the other must be vertical as well, resulting in parallel lines.

If both b1 and b2 are non-zero, then

a1b2=a2b1

implies,

a1/b1=a2/b2

implies,

-a1/b1 = -a2/b2

The lines are parallel because the slopes are equal.

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Answer 2

Answer:

Line represented by equation a1x+b1y+c1=0 and a2x + b2y+c2=0 are parallel is a1b2 =a2b1 (a₁ b₂  =  a₂b₁  )

Step-by-step explanation:

slope of equation ax + by + c = 0

m(slope ) = -b/a

a1x+b1y+c1=0  ($a_1x+b_1y+c_1=0$)

slope of above equation m₁ =  -b₁ / a₁

a2x + b2y+c2=0   ($a_2x+b_2y+c_2=0$)

slope of above equation m₂ =  -b₂ / a₁

For lines to be parallel slopes must be equal

m₁ = m₂

-b₁ / a₁ =  -b₂ / a₂

hence

a₁ b₂  =  a₂b₁  ( a1b2 =a2b1 )

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