Question

The point on the \(x\)-axis which is equidistant from the points \( (-2, 0) \) and \( (6, 0) \) is:

• A. \( (0, 2) \)
• B. \( (2, 0) \)
• C. \( (3, 0) \)
• D. \( (0, 3) \)

Hint:

On a straight line, the point equidistant from two points lies at their midpoint. Here, the midpoint of \(x=-2\) and \(x=6\) is \(x=\frac{-2+6}{2}=2\).

Answer 1

The Correct Option is B

Step 1: Represent the required point on the \(x\)-axis.
Any point on the \(x\)-axis is of the form \( (x, 0) \).
Step 2: Use the equidistant condition (distance formula).
Equidistant from \( (-2,0) \) and \( (6,0) \) means:
\[ \sqrt{(x+2)^2+(0-0)^2} \;=\; \sqrt{(x-6)^2+(0-0)^2}. \]
Step 3: Square and solve for \(x\).
\[ (x+2)^2=(x-6)^2 \;\Rightarrow\; x^2+4x+4=x^2-12x+36 \;\Rightarrow\; 16x=32 \;\Rightarrow\; x=2. \]
Thus the point is \( (2,0) \).