find the equation of a straight line which is equidistant from the points(2,3) and (6,1) expressing it in the form ax+by=c where a,b and c are constants.
Equidistant of course means that the line its perpendicular to the two lines given find the gradient first that is
Gradient=Dy/DX
1-3/6-4= -1/2
Because it is perpendicular
M1XM2 =-1
Gradient of perpendicular = 2
Equidistant now (6+2)/2. (3+1)/2
Then follow it to the end
The line must be the perpendicular bisector of the segment between the two points.
Since the slope of that segment is (1-3)/(6-2) = -1/2, your line must have slope 2
The midpoint of the segment is (4,2), so using the point-slope form, your line is
y-3 = 2(x-2)
now just put that into the desired form.
not understood
To find the equation of a straight line that is equidistant from two given points, follow these steps:
Step 1: Find the midpoint of the line segment connecting the two given points.
- Given points: (2,3) and (6,1)
- Midpoint formula: ( (x1+x2)/2 , (y1+y2)/2 )
- Midpoint coordinates: ( (2+6)/2 , (3+1)/2 )
= (4, 2)
Step 2: Find the slope of the line passing through the given points.
- Slope formula: (y2 - y1) / (x2 - x1)
- Given points: (2,3) and (6,1)
- Slope: (1 - 3) / (6 - 2)
= -2/4
= -1/2
Step 3: Find the slope of the line perpendicular to the given line.
- The slopes of perpendicular lines are negative reciprocals.
- Perpendicular slope: -1 / (-1/2)
= 2
Step 4: Use the midpoint coordinates and the perpendicular slope to find the equation of the line.
- Equation form: ax + by = c
- Using point-slope form: y - y1 = m(x - x1), where (x1, y1) is the midpoint and m is the perpendicular slope.
- Substituting the values: y - 2 = 2(x - 4)
- Expand and simplify: y - 2 = 2x - 8
2x - y = 6
Therefore, the equation of the straight line that is equidistant from the points (2,3) and (6,1) is 2x - y = 6 in the form ax + by = c, where a = 2, b = -1, and c = 6.
Why was the math book sad?
Because it had too many problems!
Now, let's find the equation of a straight line that is equidistant from the points (2,3) and (6,1).
To solve this, we first find the midpoint of the line segment connecting the two given points. The midpoint formula is given by:
(x₁ + x₂)/2 , (y₁ + y₂)/2
Using the coordinates of the points, we have:
(2 + 6)/2 , (3 + 1)/2
= 8/2 , 4/2
= 4 , 2
Now, we'll find the perpendicular bisector of the line segment. The slope of the line that connects the two given points is given by:
m = (y₂ - y₁)/(x₂ - x₁)
Using the coordinates of the points, we have:
m = (1 - 3)/(6 - 2)
= -2/4
= -1/2
The negative reciprocal of the slope will give us the slope of the perpendicular bisector. Therefore, the slope of the perpendicular bisector is:
m = 2
Now that we have the slope and the midpoint, we can use the point-slope form of a linear equation to find the equation of the perpendicular bisector:
y - y₁ = m(x - x₁)
Using the values from the midpoint (4,2), we have:
y - 2 = 2(x - 4)
Expanding the equation, we get:
y - 2 = 2x - 8
To express the equation in the form ax + by = c, we rearrange the equation:
2x - y = 6
Therefore, the equation of the straight line equidistant from the points (2,3) and (6,1) is 2x - y = 6.