Question

The point on the X-axis which is equidistant from the points (2, -5) and (-2, 9) is

• A. (-7, 0)
• B. (0, -7)
• C. (7, 0)
• D. (0, 7)

Answer 1

The Correct Option is A

Step 1: Represent the point on the X-axis.

A point on the X-axis has coordinates of the form \( (x, 0) \), since its \( y \)-coordinate is \( 0 \).

Step 2: Use the distance formula.

The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \]

Let the point on the X-axis be \( (x, 0) \). The distances from \( (x, 0) \) to \( (2, -5) \) and \( (-2, 9) \) must be equal. Thus:

\[ \sqrt{(x - 2)^2 + (0 - (-5))^2} = \sqrt{(x - (-2))^2 + (0 - 9)^2}. \]

Step 3: Simplify the equation.

Simplify both sides:

\[ \sqrt{(x - 2)^2 + 5^2} = \sqrt{(x + 2)^2 + 9^2}. \]

Square both sides to eliminate the square roots:

\[ (x - 2)^2 + 25 = (x + 2)^2 + 81. \]

Expand both sides:

\[ (x^2 - 4x + 4) + 25 = (x^2 + 4x + 4) + 81. \]

Simplify:

\[ x^2 - 4x + 29 = x^2 + 4x + 85. \]

Cancel \( x^2 \) from both sides:

\[ -4x + 29 = 4x + 85. \]

Rearrange terms:

\[ -4x - 4x = 85 - 29 \implies -8x = 56 \implies x = -7. \]

Step 4: Write the coordinates of the point.

The point on the X-axis is \( (-7, 0) \).

Final Answer: The point on the X-axis is \( \mathbf{(-7, 0)} \), which corresponds to option \( \mathbf{(1)} \).