We use the distance formula, which states that the distance between two points (x_1, y_1) and (x_2, y_2) is √((x_2 - x_1)² + (y_2 - y_1)²). We will apply this formula for PA and PB.

The coordinates of a point on the x-axis, which is equidistant from ( - 2, 5) and (2, - 3) are :

Answer: $(-2, 0)$

Step-by-step solution

Step 1: Define the point on the x-axis

Let the point on the $x$ -axis be $P$ . Since it lies on the $x$ -axis, its $y$ -coordinate must be $0$ . So, we can represent the point as $(x, 0)$ .

Step 2: Set up the distance equation

Let the given points be $A(-2, 5)$ and $B(2, -3)$ . The problem states that point $P$ is equidistant from $A$ and $B$ . This means the distance from $P$ to $A$ is equal to the distance from $P$ to $B$ .

Step 3: Apply the distance formula

We use the distance formula, which states that the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . We will apply this formula for $PA$ and $PB$ .

Step 4: Substitute coordinates and solve for x

Substitute the coordinates of $P(x, 0)$ , $A(-2, 5)$ , and $B(2, -3)$ into the distance formula. Square both sides to eliminate the square roots. Then, expand and simplify the equation to solve for $x$ . The $x^2$ terms cancel out, leading to a linear equation.

Step 5: State the coordinates of P

Since we found $x = -2$ , the coordinates of the point $P$ on the $x$ -axis that is equidistant from the given points are $(-2, 0)$ .

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