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Show that two lines a1x+b1y+c1=0and a2x+...

Show that two lines `a_1x+b_1y+c_1=0`and `a_2x+b_2y+c_2=0`, where `b_1,b_2!=0`are : (i) Parallel if `(a_1)/(b_1)=(a_2)/(b_2),`and (ii) perpendicular if `a_1a_2+b_1b_2=0`.

Text Solution

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(i)The equations of the lines are `a1​x+b1​y+c1​=0`
slope `m1​=(​a1​​) /(−b1)`
and `a2​x+b2​y+c2​=0`
Slope `m2​=(​a2​​)/(−b2)`
If the lines are parallel to each other, then
`m1​=m2​`
`⇒(a1​)/(-b1​​)=(a2​)/(-b2​​)`
...
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