Find the point (o,b) on the y-axis on the y-axis that is equidistant from the points (1,1) and (2,-1)
To find the point (o, b) on the y-axis that is equidistant from the points (1, 1) and (2, -1), we need to calculate the distance between the point (o, b) and each of the given points.
Let's start by finding the distance between the point (o, b) and the point (1, 1). The distance formula is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of the points, we have:
Distance = sqrt((1 - o)^2 + (1 - b)^2)
Similarly, let's find the distance between the point (o, b) and the point (2, -1):
Distance = sqrt((2 - o)^2 + (-1 - b)^2)
Since we want the point (o, b) to be equidistant from both points, the two distances must be equal. Therefore, we can set up an equation:
sqrt((1 - o)^2 + (1 - b)^2) = sqrt((2 - o)^2 + (-1 - b)^2)
To solve this equation, we will need to square both sides to eliminate the square root:
((1 - o)^2 + (1 - b)^2) = ((2 - o)^2 + (-1 - b)^2)
Expanding both sides, we get:
(1 - o)^2 + (1 - b)^2 = (2 - o)^2 + (-1 - b)^2
Simplifying further, we have:
1 - 2o + o^2 + 1 - 2b + b^2 = 4 - 4o + o^2 + 1 + 2b + b^2
Combining like terms, we get:
2 - 2o - 2b = 5 - 4o + 2b
Rearranging the equation, we have:
-2o - 4b = 3 - 4o
Now, let's isolate 'o' in terms of 'b':
-2o + 4o = 3 + 4b
2o = 3 + 4b
o = (3 + 4b) / 2
Since we are looking for a point (o, b) on the y-axis, the x-coordinate (o) should be 0. Therefore:
0 = (3 + 4b) / 2
Solving for 'b':
3 + 4b = 0
4b = -3
b = -3/4
So, the point (o, b) on the y-axis that is equidistant from the points (1, 1) and (2, -1) is (0, -3/4).