Question

Write an equation of the line that passes

through the given point and is (a) parallel and
(b) perpendicular to the given line. (2, 2), (1,
6) are the points of the first line, and (4, 3) is
the other point.

Answer 1

The equation for the line parallel to the one passing through points (2, 2) and (1, 6) through point (4, 3) is y = -4x + 19, and for the line perpendicular to it through the same point is y = 1/4x + 2.

Step-by-step explanation:

To write an equation of a line that is parallel or perpendicular to a given line, we must first determine the slope of the original line using two given points. For the line passing through points (2, 2) and (1, 6), the slope (m) can be calculated as follows:

m = (Y₂ - Y₁) / (X₂ - X₁)

For the points (2, 2) and (1, 6), this will be:

m = (2 - 6) / (2 - 1) = -4

(a) A line parallel to the original will have the same slope, -4. Using the point-slope form equation y - y₁ = m(x - x₁) with the point (4, 3), the parallel line's equation is:

y - 3 = -4(x - 4)

y = -4x + 16 + 3

y = -4x + 19

(b) A line perpendicular to the original will have a slope that is the negative reciprocal of -4, which is 1/4. Using the point-slope form for point (4, 3) again, the perpendicular line's equation is:

y - 3 = 1/4(x - 4)

y = 1/4x - 1 + 3

y = 1/4x + 2

Therefore, the equations for the lines parallel and perpendicular through point (4, 3) are y = -4x + 19 and y = 1/4x + 2, respectively.