Question

The coordinates of the point on the x-axis and equidistant from the points (5, -2) and (-3, 2) will be:

• A. (2, 0)
• B. (2, 2)
• C. (2, 1)
• D. (1, 0)

Hint:

When a point lies on the x-axis, its y-coordinate is always zero. Use the distance formula to equate distances for equidistant conditions.

Answer 1

The Correct Option is A

Step 1: Concept.
The required point lies on the x-axis, so its coordinates are \( (x, 0) \). It is equidistant from the two points \( (5, -2) \) and \( (-3, 2) \).
Step 2: Use the distance formula.
Equidistant means: \[ \sqrt{(x - 5)^2 + (0 + 2)^2} = \sqrt{(x + 3)^2 + (0 - 2)^2} \]
Step 3: Square both sides.
\[ (x - 5)^2 + 4 = (x + 3)^2 + 4 \] Simplify: \[ x^2 - 10x + 25 = x^2 + 6x + 9 \]
Step 4: Simplify further.
\[ -10x + 25 = 6x + 9 \Rightarrow 16x = 16 \Rightarrow x = 1 \] Step 5: Final coordinates.
Since the point lies on the x-axis, the coordinates are \( (1, 0) \). Step 6: Verify options.
Option (D) (1, 0) is correct.