Find coordinates of the point, which when joined with #(c_1,c_2)# forms a line that is parallel to the line joining #(a_1,a_2)# and #(b_1,b_2)#?

1 Answer
Shwetank Mauria
Sep 26, 2016

Any point lying on #(b_2-b_1)(x-c_1)-(y-c_2)(a_2-a_1)=0# when joined with #(c_1,c_2)# would form a line parallel to the line joining #(a_1,a_2)# and #(b_1,b_2)#.

Explanation:

The slope of a line passing through #(a_1,a_2)# and #(b_1,b_2)# is #(b_2-b_1)/(a_2-a_1)#

The equation of a line with a slope #m# and passing through #(x_1,y_1)# is #y-y_1=m(x-x_1)#

As the slope of the line parallel to above too would be #(b_2-b_1)/(a_2-a_1)# and as it passes through #(c_1,c_2)#, its equation would be

#y-c_2=(b_2-b_1)/(a_2-a_1)(x-c_1)# or

#(b_2-b_1)(x-c_1)-(y-c_2)(a_2-a_1)=0#

Hence any point lying on #(b_2-b_1)(x-c_1)-(y-c_2)(a_2-a_1)=0# will satisfy the given condition, i.e. when joined with #(c_1,c_2)# would form a line parallel to the line joining #(a_1,a_2)# and #(b_1,b_2)#.

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