Question

The equation of the line parallel to the line \[2x - 3y = 1\] and passing through the middle point of the line segment joining the points \[(1,3)\] and \[(1, - 7)\], is
1) \[2x - 3y + 8 = 0\]
2) \[2x - 3y = 8\]
3) \[2x - 3y + 4 = 0\]
4) \[2x - 3y = 4\]

Answer 1

Hint: we can solve the given simple problem by using the concepts of equations of parallel line and the midpoint formula. Since we have to find the equation of the line which is parallel to the line \[2x - 3y = 1\] so we have to assume the equation of the parallel line by keeping the terms containing x and y unaltered and change the constant. The constant k can be determined by using the next condition given in the problem, that is by using the midpoint of \[(1,3)\] and \[(1, - 7)\] given in the problem.

Complete step by step answer:
Let us consider the given equation of line \[2x - 3y = 1\]
We can write the above equation as \[2x - 3y - 1 = 0\]
Since we have to find the equation of line parallel to the above line so
let us assume the equation of parallel line will be of the form \[2x - 3y - k = 0 - - - \left( 1 \right)\]
now we need to find the value of k, since it is given that the line passing through the middle point of the line segment joining the points \[(1,3)\] and \[(1, - 7)\], therefore let us find the midpoint \[(x,y)\] by using the midpoint formula given by
\[{\text{midpoint = }}\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\]
let \[({x_1},{y_1}) = (1,3)\] and \[({x_2},{y_2}) = (1, - 7)\]
on substitution we get \[{\text{(x,y) = }}\left( {\dfrac{{1 + 1}}{2},\dfrac{{3 - 7}}{2}} \right)\]
in simplification we get \[{\text{(x,y) = }}\left( {1, - 2} \right)\]
now substitute these x and y values in equation 1 we get
\[2(1) - 3( - 2) - k = 0\]
\[k = 2 + 6\]
\[k = 8\]
Now again substitute k value in equation 1 we get required line equation
\[2x - 3y - 8 = 0\]
\[2x - 3y = 8\]

So, the correct answer is “Option 2”.

Note: Midpoint formula is used to find the center point of a straight line. Sometimes you will need to find the number that is half of two particular numbers. ... In that similar fashion, we use the midpoint formula in coordinate geometry to find the halfway number (i.e., point) of two coordinates.