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Prove that distance between two parallel...

Prove that distance between two parallel lines `ax+by+c_1=0` and `ax+by+c_2=0` is given by `|c_1-c_2|/(sqrt(a^2+b^2))`.

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Distance between the parallel lines ax + by +c_1 = 0 and ax + by + c_2 = 0 is

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Knowledge Check

  • Find the distance between the parallel lines ax+by+ c=0 and ax +by + d=0 .

    A
    `| (d-c)/( sqrt(a^2 + b^2 ) )|`
    B
    `| (d-c)/( sqrt(a^2 - b^2 ) )|`
    C
    `(d-c)/( sqrt(ab)`
    D
    `d-c`

  • Consider the following statements : I . The distance between the lines y = mx + c_(1) and y = mx + c_(2) is (|c_(1) - c_(2)|)/(sqrt(1 -m^(2))) II . The distance between the lines ax + by + c_(1) = 0 and ax + by + c_(2) = 0 is (|c_(1) - c_2|)/(sqrt(a^(2) + b^(2))) III . The distance between the lines x = c and x = c_(2) is |c_(1) - c_(2)| Which of the above statements are correct ?

    A
    I and II only
    B
    II and III only
    C
    I and III only
    D
    I , II and III only

  • Consider the following statements : 1. The distance between the lines y=mx+c_(1) and y=mx+c_(2) is (|c_(1)-c_(2)|)/sqrt(1-m^(2)) . 2. The distance between the lines ax+by+c_(1) and ax+by+c_(2)=0 is (|c_(1)-c_(2)|)/sqrt(a^(2)+b^(2)) . 3. The distance between the lines x=c and x=c_(2) is |c_(1)-c_(2)| . Which of the above statements are correct ?

    A
    1 and 2 only
    B
    2 and 3 only
    C
    1 and 3 only
    D
    1, 2 and 3

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    For a>b>c>0, if the distance between (1,1) and the point of intersection of the line ax+by-c=0 and bx+ay+c=0 is less than 2sqrt(2) then,(A)a+b-c>0(B)a-b+c 0(D)a+b-c<0

    For a gt b gt c gt 0 if the distance between (1,1) and the point of intersection of the lines ax + by +c=0 and bx + ay+c=0 is less than 2sqrt2 then

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