Loci, Spheres and Cuboid Problems: Quick Revision

A brief note on Loci, Spheres and Cuboid Problems. Read it before a test. Try each question at the end before you open its answer.

The sphere

• Centre $(a, b, c)$, radius $r$: $(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$.
• General form: $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$. Centre $(-u, -v, -w)$, $r^2 = u^2 + v^2 + w^2 - d$.
• Diameter $AB$: the centre is the midpoint and the radius is half of $AB$. Or $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) + (z - z_1)(z - z_2) = 0$.
• A plane $z = h$ cuts a circle of radius $\sqrt{r^2 - (h - c)^2}$. The $x$-axis is cut in a chord of length $2\sqrt{r^2 - b^2 - c^2}$.

Loci from two points

• $PA = PB$ and $PA^2 - PB^2 = k$: a plane. The squares cancel.
• $PA^2 + PB^2 = k$ and $PA = \lambda PB$ with $\lambda \ne 1$: a sphere.
• Four points not in one plane: one point is equally far from all. Solve three linear equations.

Cubes, cuboids and planes

• Cube of side $a$: face diagonal $a\sqrt{2}$, body diagonal $a\sqrt{3}$, centre $\left(\dfrac{a}{2}, \dfrac{a}{2}, \dfrac{a}{2}\right)$.
• Cuboid: diagonal $\sqrt{a^2 + b^2 + c^2}$.
• Nearest point of a line: write $d^2$ as a quadratic in $t$ and complete the square.
• Plane $x = k$ divides $AB$ in the ratio $(k - x_1) : (x_2 - k)$.

A worked example

Find the locus of $P$ with $PA = PB$ for $A(0, 0, 0)$ and $B(2, 2, 1)$.

1. Square. $x^2 + y^2 + z^2 = (x - 2)^2 + (y - 2)^2 + (z - 1)^2$.
2. Simplify. The squares cancel: $0 = -4x - 4y - 2z + 9$, so $4x + 4y + 2z = 9$.
3. Check. The midpoint $\left(1, 1, \dfrac{1}{2}\right)$ gives $4 + 4 + 1 = 9$, so it lies on the plane.

Practice with new numbers

A program to try

The program checks the main results on small examples.

Mistakes to avoid

• Reading the centre with the wrong signs. In $x^2 + 2ux$ the centre has $x = -u$.
• Forgetting the root. $r^2 = u^2 + v^2 + w^2 - d$, then take the square root.
• Using $a\sqrt{2}$ for a body diagonal. The body diagonal of a cube is $a\sqrt{3}$.
• Expecting a line for the points equally far from two points. In space it is a plane.

Check yourself

1. Find the centre and radius of $x^2 + y^2 + z^2 - 4x + 2y - 6z + 5 = 0$.

2. Find the locus of the points equally far from $(0, 0, 0)$ and $(0, 0, 6)$.

3. Find the diagonal of a cuboid with edges $2$, $3$ and $6$.

4. In what ratio does the plane $x = 3$ divide the join of $(1, 0, 0)$ and $(9, 4, 4)$?

5. What is the least value of $d^2$ from $(a, 0, 0)$ to the line $(t, t, t)$ for $a = 3$?

Sources

• NCERT, Mathematics Textbook for Class XI, the chapter Introduction to Three Dimensional Geometry.
• OpenStax, Calculus Volume 3, Chapter 2, section 2.2 on vectors in three dimensions.