Question

The coordinate of the point on the $x$-axis and equidistant from the points $(2, -5)$ and $(-2, 9)$ will be:

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

Hint:

If a point lies on the $x$-axis, its $y$-coordinate is always $0$. Use the distance formula and simplify carefully to find the $x$-coordinate.

Answer 1

The Correct Option is A

Step 1: Assume the coordinates of the required point.
Since the point lies on the $x$-axis, its coordinates will be $(x, 0)$.

Step 2: Write the distance formula.
If a point $(x, 0)$ is equidistant from $(2, -5)$ and $(-2, 9)$, then: \[ \text{Distance from } (x, 0) \text{ to } (2, -5) = \text{Distance from } (x, 0) \text{ to } (-2, 9) \] \[ \sqrt{(x - 2)^2 + (0 + 5)^2} = \sqrt{(x + 2)^2 + (0 - 9)^2} \]
Step 3: Simplify by squaring both sides.
\[ (x - 2)^2 + 25 = (x + 2)^2 + 81 \] \[ x^2 - 4x + 4 + 25 = x^2 + 4x + 4 + 81 \]
Step 4: Simplify further.
\[ -4x + 29 = 4x + 85 \] \[ 8x = -56 \] \[ x = -7 \]
Step 5: Write the coordinates.
Since the point lies on the $x$-axis, its coordinates are $(-7, 0)$.

Step 6: Conclusion.
The required point on the $x$-axis, equidistant from $(2, -5)$ and $(-2, 9)$, is $(-7, 0)$.
Final Answer
Final Answer: \[ \boxed{(-7, 0)} \]