Question

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)`?

Answer 1

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)`.