The (x,y) is equidistant from (0,0) and (4,-2). Find its algebraic equation.
must be on the right bisector of the line joining (0,0) and 4,-2)
slope of that line is (-2-0)/(4-0) = -1/2
so the slope of the right-bisector must be +2
and we can start by saying the equation of the right-bisector must be
y = 2x + b
but we also know that the mid point of our line segment must lie on this
midpoint = ((4+0)/2 , (-2+0)/2 ) = (2, -1)
so in y = mx + b
-1 = 3(2) + b
b = -7
y = 2x - 7
To find the equation of a point that is equidistant from two given points, you can use the midpoint formula. Let's call the coordinates of the point (x, y). Using the midpoint formula, the x-coordinate of the midpoint is the average of the x-coordinates of the two given points, and the y-coordinate of the midpoint is the average of the y-coordinates of the two given points.
Given points: (0, 0) and (4, -2)
Midpoint coordinates: ((0+4)/2, (0+(-2))/2)
Calculating the midpoint coordinates:
x-coordinate of midpoint = (0 + 4) / 2 = 4 / 2 = 2
y-coordinate of midpoint = (0 + (-2)) / 2 = -2 / 2 = -1
So, the coordinates of the midpoint are (2, -1).
Now we can use the point-slope form of a linear equation to find the equation of the line passing through the midpoint and (x, y).
Point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is the midpoint and m is the slope of the line.
Since the midpoint and (x, y) are equidistant from (0, 0) and (4, -2), the line passing through them will be perpendicular to the line connecting (0, 0) and (4, -2).
To find the slope of the line connecting (0, 0) and (4, -2), we can use the slope formula: m = (y₂ - y₁) / (x₂ - x₁).
Using the coordinates of the given points:
m = (-2 - 0) / (4 - 0) = -2 / 4 = -1/2
Now, since the line passing through the midpoint and (x, y) is perpendicular, the slope of that line will be the negative reciprocal of -1/2, which is 2.
Therefore, we have:
m = 2 (slope of the line passing through the midpoint and (x, y))
(x₁, y₁) = (2, -1) (midpoint coordinates)
Using the point-slope form, we can substitute the values and simplify the equation:
y - (-1) = 2(x - 2)
y + 1 = 2x - 4
y = 2x - 5
Hence, the algebraic equation of the point (x, y) equidistant from (0, 0) and (4, -2) is y = 2x - 5.