The Locus of Points Equation

To find the equation of the locus of all points such that the difference of their distance from (4,0) and (-4,0) is always equal to 2, we can follow a step-by-step approach.

Step 1: Understanding the Problem

Before we proceed with the solution, let's understand the problem statement. We need to find the equation of the locus, which is the set of all points that satisfy a given condition. In this case, the condition is that the difference of the distances of any point from (4,0) and (-4,0) is always equal to 2.

Step 2: Distance Formula

To find the distance between two points, we can use the distance formula:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.

Step 3: Formulating the Condition

Let's consider a point (x, y) on the locus.

The distance from (x, y) to (4, 0) can be calculated as:

Distance₁ = √[(x - 4)² + (y - 0)²]

Similarly, the distance from (x, y) to (-4, 0) can be calculated as:

Distance₂ = √[(x - (-4))² + (y - 0)²]

According to the given condition, the difference between these two distances is always equal to 2.

Difference = Distance₁ - Distance₂ = 2

Step 4: Simplifying the Equation

Let's square both sides of the equation to eliminate the square roots:

(Distance₁ - Distance₂)² = 2²

Simplifying further, we get:

(Distance₁ - Distance₂)² = 4

Expanding the square, we have:

Distance₁² - 2 * Distance₁ * Distance₂ + Distance₂² = 4

Substituting the distance formulas, we get:

[(x - 4)² + y²] - 2 * [(x + 4)² + y²] + [(x + 4)² + y²] = 4

Simplifying the equation, we get:

(x - 4)² + y² - 2(x + 4)² - 2y² + (x + 4)² + y² = 4

Simplifying further, we have:

(x - 4)² - 2(x + 4)² + (x + 4)² = 4

Step 5: Final Equation

Expanding and simplifying the equation, we get:

x² - 8x + 16 - 2(x² + 8x + 16) + (x² + 8x + 16) = 4

Simplifying further, we have:

x² - 8x + 16 - 2x² - 16x - 32 + x² + 8x + 16 = 4

Combining like