Find the equation of the set of all points P(x, y) if the sum of the distances of P(x, y) from (0, -2) and (0, 2) is equal to 8.
To find the equation for the set of all points satisfying the given condition, let's break down the problem step by step.
Step 1: Understand the problem.
The problem involves finding the equation for the set of all points whose sum of distances from two given points is equal to a constant value.
Step 2: Visualize the scenario.
The two given points in this problem are (0, -2) and (0, 2). To understand the scenario, let's plot these points on a coordinate plane.
| .
2 | P .
| .
| .
0 | .
| .
-2 | .
| .
-2 0 2 4 6 8
Step 3: Identify the constant value.
According to the problem, the sum of distances from the two points should be equal to 8.
Step 4: Define variables and distances.
Let's assume the coordinates of point P as (x, y). To find the equation, let's denote the distance from P to (0, -2) as d1 and the distance from P to (0, 2) as d2.
Step 5: Apply the distance formula.
Using the distance formula, we can find d1 and d2 as follows:
d1 = sqrt((x - 0)^2 + (y - (-2))^2)
= sqrt(x^2 + (y + 2)^2)
d2 = sqrt((x - 0)^2 + (y - 2)^2)
= sqrt(x^2 + (y - 2)^2)
Step 6: Set up the equation.
According to the problem, the sum of the distances d1 and d2 should be equal to 8:
d1 + d2 = 8
Substituting the values of d1 and d2, we get:
sqrt(x^2 + (y + 2)^2) + sqrt(x^2 + (y - 2)^2) = 8
This equation represents the set of all points (x, y) satisfying the given condition.