Question

Find the equation of the line that is parallel to 2x + 5y − 7 = 0 and passes through the mid-point of the line segment joining the points (2, 7) and (−4, 1).

Answer 1

⇒ Rewriting the given equation 2x + 5y − 7 = 0 into the slope-intercept form, y = mx + c to find the slope (m₁):

5y = −2x + 7

`y = - 2/5 x + 7/5`

∴ The slope (m₁) of the given line is `- 2/5`.

⇒ Since the line is parallel, the slope (m₂) is the same:

`m_2 = - 2/5`

⇒ The line passes through the midpoint of (2, 7) and (−4, 1),

Using the mid-point formula:

`M = ((x_1 + x_2)/2, (y_1 + y_2)/2)`

`M = ((2 + (-4))/2, (7 + 1)/2)`

`M = ((-2)/2, 8/2)`

∴ M = (−1, 4)

⇒ Using the point-slope formula with `m = - 2/5` and point (−1, 4):

y − y₁ = m(x − x₁)

`y - 4 = - 2/5 (x - (-1))`

`y - 4 = - 2/5 (x + 1)`

5(y − 4) = −2(x + 1)   ...[Multiplied the entire equation by 5.]

5y − 20 = −2x − 2

⇒ Rearranging the above equation in the standard form (Ax + By + C = 0),

2x + 5y − 18 = 0

Hence, the equation of the line is 2x + 5y − 18 = 0.