Question

The locus of a point equidistant from three collinear points is:

• A. A straight line
• B. A pair of points
• C. A point
• D. The null set

Answer 1

The correct option is D The null set

Let three collinear points be (a,0), (b,0) and (0,0).
Let the point (x,y) be equidistant from these collinear points.
$\Rightarrow {(x-a)}^2+y^2=(x-b{)}^2+y^2=x^2+y^2\Rightarrow a^2-2ax=b^2-2bx=0\Rightarrow x=\frac{a}{2}=\frac{b}{2}\Rightarrow a=b$

This contradicts the uniqueness of the points. Thus, no such $(x,y)$ exists.
Option D is answer.