Show that the distance between the parallel lines `ax+by+c=0` and `k(ax+by)+d=0` is `|(c-(d)/(k))/(sqrt(a^(2)+b^(2)))|`

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

distance between the parallel planes ax+by+ cz + d =0 and ax+by + cz + d' =0 is

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))

Assertion: The distance between two parallel planes ax+by+cz+d=0 and ax+by+cz+d\'=0 is (|d-d\'|)/sqrt(a^2+b^2+c^2) , Reason: The normal of two parallel planes are perpendicular to each other. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not te correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

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_(1) and x=c_(2) is |c_(1)-c_(2)| . Which of the above statements are correct ?

Show that the equation to the parallel line mid-way between the parallel lines ax+by+c_(1)=0andax+by+c_(2)=0 is ax+by+(c_(1)+c_(2))/2=0

Statement-1 The distance between the planes 4x-5y+3z=5 and 4x-5y+3z+2=0 is (3)/(5sqrt(2)) . Statement-2 The distance between ax+by+cz+d_1=0 and ax+by+cz+d_2=0 is |(d_1-d_2)/(sqrt(a^2+b^2+c^2))| .

Two parallel lines lying in the same quadrant make intercepts a and b on x and y axes, respectively, between them. The distance between the lines is (a) (ab)/sqrt(a^2+b^2) (b) sqrt(a^2+b^2) (c) 1/sqrt(a^2+b^2) (d) 1/a^2+1/b^2

For a gt b gt c gt 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 2sqrt2 then, (A) a+b-cgt0 (B) a-b+clt0 (C) a-b+cgt0 (D) a+b-clt0

Show that the equation to the plane containing the line y/b+ z/c =1, x=0, and parallel to the line x/a -z/c=1, y=0is x/a-y/b -z/c+1=0, and if 2d is the shortest, distance between given lines, prove that (1)/(d ^(2)) = (1)/(a ^(2)) + (1)/(b ^(2)) + (1)/(c ^(2)).