Find the equation of a straight line that is equidistant from the points (-2, -1)and (4, 5) expressing it in the form ax+by=c, where a, b and c are integers
Mr. Bot fails again!!!
The equation you are looking for must be the right bisector of the
segment joining the two given points.
slope for the given points = (5+1)/(4+2) = 6/6 = 1
so the slope of the required line is -1
it is y = -x + b, but it must also go through the midpoint of the above segment
which would be (1,2) , so
2 = -1 + b, or b = 3
new equation: y = -x + 3 or
x + y = 3
or, use the point-slope form of the line to start with
slope = -1
it goes through the midpoint of AB: (1,2)
so the equation is
y-2 = -1(x-1)
which resolves to
x+y = 3
To find the equation of a line that is equidistant from two points, we can use the concept of the perpendicular bisector. The perpendicular bisector is a line that passes through the midpoint of the line segment joining the two given points and is perpendicular to that line segment.
Let's start by finding the midpoint of the line segment joining the two given points (-2, -1) and (4, 5).
Midpoint formula:
The midpoint (x, y) of a line segment with endpoints (x1, y1) and (x2, y2) can be found using the following formulas:
x = (x1 + x2) / 2
y = (y1 + y2) / 2
Using the coordinates (-2, -1) and (4, 5):
x = (-2 + 4) / 2 = 2 / 2 = 1
y = (-1 + 5) / 2 = 4 / 2 = 2
So, the midpoint of the line segment is (1, 2).
Next, we need to find the slope of the line passing through the two given points. The slope (m) of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Using the coordinates (-2, -1) and (4, 5):
m = (5 - (-1)) / (4 - (-2))
= (5 + 1) / (4 + 2)
= 6 / 6
= 1
Now, we know the slope of the line passing through the two given points is 1.
The slope of the perpendicular bisector is the negative reciprocal of the slope of the given line. Therefore, the slope of the perpendicular bisector is -1.
Using the midpoint (1, 2) and the slope -1, we can find the equation of the perpendicular bisector using the point-slope form of a line.
Point-slope form:
For a line with slope m passing through a point (x1, y1), the equation can be written as:
y - y1 = m(x - x1)
Substituting the values, we have:
y - 2 = -1(x - 1)
Simplifying the equation:
y - 2 = -x + 1
Rearranging the equation in the form ax + by = c:
x + y = 3
So, the equation of the straight line that is equidistant from the points (-2, -1) and (4, 5) is x + y = 3 in the form ax + by = c, where a, b, and c are integers.