LemonmathSelf-study sheet

Grade 12 · Lesson 5.4

The equation of a sphere

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Goals

Write the equation $(x-a)^2+(y-b)^2+(z-c)^2=R^2$ of a sphere with a given centre and radius.
Recognise when $x^2+y^2+z^2-2ax-2by-2cz+d=0$ is a sphere and find its centre and radius.
Write the equation of a sphere with a given diameter, through given points, or tangent to a plane.
Decide where a point or a plane lies relative to a sphere; write the equation of a tangent plane.

Remember

Centre $I(a, b, c)$ , radius $R$ : $(x-a)^2+(y-b)^2+(z-c)^2=R^2$ .
$x^2+y^2+z^2-2ax-2by-2cz+d=0$ is a sphere $\iff a^2+b^2+c^2-d>0$ ; then $R=\sqrt{a^2+b^2+c^2-d}$ .
Diameter $AB$ : the centre is the midpoint of $AB$ and $R=\dfrac{AB}{2}$ .
A point $M$ : $IM<R$ inside, $IM=R$ on, $IM>R$ outside the sphere.
A plane: $h=d(I, (P))$ ; $h>R$ no common point, $h=R$ tangent, $h<R$ a circle of radius $\sqrt{R^2-h^2}$ .
The tangent plane at $M$ : through $M$ with normal vector $\overrightarrow{IM}$ .

Exercises

Exercise 1

Write the equation of the sphere with centre $I(-1, 2, 0)$ and radius $3$ . Is the point $A(1, 3, 2)$ inside, on or outside the sphere?

Exercise 2

Find the centre and radius of the sphere $x^2+y^2+z^2+4x-6y+2z-11=0$ .

Exercise 3

Which of these are equations of spheres? Find the centre and radius of each sphere.

a) $x^2+y^2+z^2-2x+4z+1=0$ . b) $x^2+y^2+z^2+2y+5=0$ . c) $2x^2+2y^2+2z^2-4x+8y-2=0$ .

Exercise 4

Write the equation of the sphere with diameter $AB$ , where $A(2, -1, 3)$ and $B(0, 3, -1)$ .

Exercise 5

(Revisiting The equation of a plane.) Write the equation of the sphere with centre $I(1, 2, -1)$ that touches the plane $(P): 2x-2y+z+9=0$ .

Exercise 6

Let $(S): (x-2)^2+(y+1)^2+z^2=9$ . Is each claim true or false?

a) $(S)$ has centre $I(2, -1, 0)$ .

b) $(S)$ has radius $9$ .

c) The point $M(4, 1, 1)$ lies on $(S)$ .

d) The plane $z=4$ cuts $(S)$ in a circle.

Exercise 7

A Bluetooth speaker hangs at $I(2, 3, 4)$ in a room (unit: metre), and the floor is the plane $z=0$ . The speaker can be heard clearly within $6$ m of it. Find the area of the part of the floor where the speaker is heard clearly (in m², to 1 decimal place).

Exercise 8

Write the equation of the sphere whose centre lies on the axis $Ox$ and which passes through $A(1, 2, 3)$ and $B(3, 0, 1)$ .

Exercise 9

(Revisiting The equation of a line in space.) Find where the line $d: x=1+t$ , $y=2$ , $z=2-3t$ meets the sphere $(S): x^2+y^2+z^2=9$ , and find the length of the chord.

Exercise 10 Challenge

Model the Earth as a sphere with centre $O$ and radius $6400$ km. With one thousand km as the unit: $(S): x^2+y^2+z^2=40.96$ . A satellite is at $V(0, 0, 8)$ . The points $M$ on the ground that see the satellite on the horizon are those for which $VM$ is tangent to the sphere, that is $\overrightarrow{OM}\perp\overrightarrow{MV}$ . Prove that these points lie in one plane and find the radius of the circle bounding the coverage area (in km).

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. In the system

Oxyz

, the sphere

(x+1)^2+(y-2)^2+z^2=4

has centre and radius

I(1, -2, 0)

R=2

.B.

I(-1, 2, 0)

R=2

.C.

I(-1, 2, 0)

R=4

.D.

I(1, -2, 0)

R=4

Question 2. The radius of the sphere

x^2+y^2+z^2-2x+4y-6z+5=0

9

.B.

\sqrt{19}

.C.

3

.D.

\sqrt{14}

Question 3. Which of these is the equation of a sphere?

x^2+y^2+z^2-2x+4y+6=0

.B.

x^2+y^2+z^2-4x+2z+1=0

.C.

x^2+2y^2+z^2-2=0

.D.

x^2+y^2-z^2+2x=0

Question 4. The sphere with centre

I(1, 0, -2)

that touches the plane

(P): 2x-y+2z+8=0

has equation

(x-1)^2+y^2+(z+2)^2=4

.B.

(x-1)^2+y^2+(z+2)^2=2

.C.

(x+1)^2+y^2+(z-2)^2=4

.D.

(x-1)^2+y^2+(z+2)^2=36

Part II. True or false

Question 1. Let

(S): x^2+y^2+z^2-2x-4y+4z=0

a) $(S)$ has centre $I(1, 2, -2)$ .	T / F
b) $(S)$ has radius $9$ .	T / F
c) The origin $O$ lies on $(S)$ .	T / F
d) The plane $(P): x+2y-2z-18=0$ touches $(S)$ .	T / F

Part III. Short answer

Question 1. What is the radius of the sphere with diameter

AB

, where

A(1, 2, 3)

and

B(3, -2, -1)

Question 2. The plane

z=1

cuts the sphere

(x-1)^2+(y+2)^2+(z-4)^2=25

in a circle. Find the area of the disc bounded by that circle (to 1 decimal place).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $(x-a)^2+(y-b)^2+(z-c)^2=R^2$ ; compare $IA$ with $R$ .

Steps:

Substitute $a=-1$ , $b=2$ , $c=0$ , $R=3$ .
Compute $IA^2$ .

Exercise 2.

What you need: $a$ , $b$ , $c$ are the coefficients of $x$ , $y$ , $z$ divided by $-2$ ; $R=\sqrt{a^2+b^2+c^2-d}$ .

Steps:

Read off $a$ , $b$ , $c$ , $d$ .
Compute $R$ .

Exercise 3.

What you need: the condition $a^2+b^2+c^2-d>0$ , with the coefficients of $x^2$ , $y^2$ , $z^2$ equal to $1$ .

Steps:

In part c), divide both sides by $2$ first.
Check the condition for each equation.

Exercise 4.

What you need: the centre is the midpoint of $AB$ and $R=\dfrac{AB}{2}$ .

Steps:

Find the midpoint $I$ .
Find $AB$ , then $R$ .

Exercise 5.

What you need: the sphere touches $(P)$ when $R=d(I, (P))$ .

Steps:

Find $d(I, (P))$ with the distance formula.
Write the equation with this $R$ .

Exercise 6.

What you need: read off the centre and radius; compare the distance from the centre to the plane with $R$ .

Steps:

Read off $I$ and $R$ .
Substitute $M$ into the left-hand side.
Find the distance from $I$ to the plane $z=4$ .

Exercise 7.

What you need: a plane at distance $h<R$ from the centre cuts the sphere in a circle of radius $r=\sqrt{R^2-h^2}$ .

Steps:

$h=4$ , $R=6$ ; find $r$ .
The area of a disc is $\pi r^2$ .

Exercise 8.

What you need: a centre on $Ox$ has the form $I(t, 0, 0)$ ; the condition $IA=IB$ .

Steps:

Write $IA^2=IB^2$ in terms of $t$ .
Solve for $t$ , then find $R^2=IA^2$ .

Exercise 9.

What you need: substituting the parametric equations into the equation of the sphere gives a quadratic equation in $t$ .

Steps:

Substitute $x$ , $y$ , $z$ into $(S)$ and simplify.
Solve for $t$ , find the two points, then the distance between them.

Exercise 10.

What you need: $\overrightarrow{OM}\cdot\overrightarrow{MV}=0$ ; $OM^2=40.96$ ; $r=\sqrt{R^2-h^2}$ .

Steps:

Write $\overrightarrow{MV}=\overrightarrow{OV}-\overrightarrow{OM}$ and expand $\overrightarrow{OM}\cdot\overrightarrow{MV}=0$ .
Deduce $8z=40.96$ , then find $r$ .

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