Find the equation of the set of all points P(x, y) which is equidistant from (2, -4) and (-1, 5).
(x – 2)^2 + (y + 4)^2 = (x + 1)^2 + (y – 5)^2
x^2 – 4x + y^2 + 8y + 20 = x^2 + 2x + y^2 – 10y + 26
x^2 – 4x + y^2 + 8y + 20 – (y^2 + 8y + 20) = x^2 + 2x + y^2 – 10y + 26 – (y^2 + 8y + 20)
x^2 – 4x – (x^2 + 2x) = x^2 + 2x – 18y + 6 – (x^2 + 2x)
-6x = -18y + 6
(-6x) / (-6) = (-18y + 6) / (-6)
x = 3y – 1
x – 3y + 1 = 0
To find the equation of the set of all points equidistant from two given points, you can use the concept of midpoint and distance formula.
Let's call the given points A(2, -4) and B(-1, 5). Now, we need to find the equation of all points P(x, y) that are equidistant from both A and B.
Step 1: Find the midpoint M between the two given points using the midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
=> M = ((2 + (-1))/2, ((-4) + 5)/2)
=> M = (1/2, 1/2)
Step 2: Calculate the distance between one of the given points (A or B) and the midpoint (M) using the distance formula: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For point A(2, -4):
=> Distance_A = sqrt((1/2 - 2)^2 + (1/2 - (-4))^2)
Simplifying further, we get:
=> Distance_A = sqrt((1/2 - 4/2)^2 + (1/2 + 8/2)^2)
=> Distance_A = sqrt((-3/2)^2 + (9/2)^2)
=> Distance_A = sqrt(9/4 + 81/4)
=> Distance_A = sqrt(90/4)
=> Distance_A = sqrt(90)/2
=> Distance_A = (3sqrt(10))/2
Similarly, for point B(-1, 5):
=> Distance_B = sqrt((1/2 - (-1))^2 + (1/2 - 5)^2)
Simplifying further, we get:
=> Distance_B = sqrt((3/2)^2 + (-9/2)^2)
=> Distance_B = sqrt(9/4 + 81/4)
=> Distance_B = sqrt(90/4)
=> Distance_B = sqrt(90)/2
=> Distance_B = (3sqrt(10))/2
Step 3: Equate the distances between each point (P) and the midpoint (M) to find the equation:
The equation will be:
Distance_MP = sqrt((x - 1/2)^2 + (y - 1/2)^2) = (3sqrt(10))/2
To simplify further, we can square both sides of the equation to eliminate the square root:
(x - 1/2)^2 + (y - 1/2)^2 = (3sqrt(10)/2)^2
(x - 1/2)^2 + (y - 1/2)^2 = (9*10)/4
(x - 1/2)^2 + (y - 1/2)^2 = 90/4
(x - 1/2)^2 + (y - 1/2)^2 = 45/2
Therefore, the equation of the set of all points equidistant from (2, -4) and (-1, 5) is (x - 1/2)^2 + (y - 1/2)^2 = 45/2.