Q.Find the equation of the set of points which are equidistant from the points $(1, 2, 3)$ and $(3, 2, -1)$.

Concept: 3D Coordinate Geometry — Section Formula / Locus of equidistant points (perpendicular bisector plane).

Let $P(x, y, z)$ be any point equidistant from $A(1, 2, 3)$ and $B(3, 2, -1)$.

Step 1: Equate distances $PA = PB$ using the distance formula:

Step 2: Square both sides and cancel $(y-2)^2$:

Step 3: Expand and simplify: …

Why this works — the 3D coordinate octants and the idea of a locus

In three-dimensional space, the condition “equidistant from two given points” defines a plane. Why? Because in any geometry (2D or 3D), the set of points at equal distance from two fixed points is the perpendicular bisector of the segment joining them. In 2D, that’s a line; in 3D, it’s a plane — the plane that cuts the segment at its midpoint and stands perpendicular to it.

Think of it this way: if you have two points $A$ and $B$, any point $P$ that is equally far from $A$ and $B$ must lie on the “middle surface” that is orthogonal to $AB$ and passes through its midpoint. That surface is flat — a plane.

So our job is to find the equation of that plane. We don’t need to visualise octants here; we just use the distance formula and simplify.

Step-by-step solution

1. Write the distance condition algebraically

Let $P(x, y, z)$ be any point satisfying the condition. The distance from $P$ to $A(1,2,3)$ equals the distance from $P$ to $B(3,2,-1)$.

Using the 3D distance formula:

2. Square both sides to remove the square roots

Since both sides are non-negative, squaring is safe:

Notice that $(y-2)^2$ appears on both sides — they cancel immediately.

3. Expand the remaining squares

Left side: $(x-1)^2 + (z-3)^2 = (x^2 - 2x + 1) + (z^2 - 6z + 9)$

Right side: $(x-3)^2 + (z+1)^2 = (x^2 - 6x + 9) + (z^2 + 2z + 1)$

4. Cancel common terms

Both sides have $x^2$ and $z^2$. Subtract them from both sides:

Simplify constants: $1+9 = 10$ on left, $9+1 = 10$ on right. So:

Cancel $10$ from both sides:

5. Collect terms to one side …