Question

Hi can someone help me determine how to solve this ?
Determine an equation of a sphere given that one of the diameters has the extremities: (2,1,4) and (4,3,10)

Answer 1

What does extremities mean?
From one end to the other? Like the diameter or something? 0_o

Answer 2

I'm not sure ... It's translated from french

Answer 3

I have the answer though if you want

Answer 4

Oh sure :)

Answer 5

I just don't know how to get it loll

Answer 6

\[(x-3)^{2}+(y -2)^{2}+(z-7)^2 = 11\]

Answer 7

Oo nice I got the right answer :)
Ok let's see if I can explain this in a way that makes sense.

Answer 8

Nice.. All i know is finding the radius .. but after i'm not sure what to do

Answer 9

So we'll start by finding the `diameter`.
They gave us the coordinates from one end of the sphere to the other.

Using the distance formula will give us the diameter.

Answer 10

sqrt of 44?

Answer 11

\[\Large\bf\sf (2,1,4),\qquad (4,3,10)\]
\[\Large\bf\sf d\quad=\quad \sqrt{(4-2)^2+(3-1)^2+(10-4)^2}\]

Answer 12

Yes good good.
Let's simplify that value a lil bit.

Answer 13

\[\Large\bf\sf \sqrt{44}\quad=\quad \sqrt{4\cdot11}\]

Answer 14

We have a perfect square we can pull out of the root, yes?

Answer 15

Agreed

Answer 16

\[\Large\bf\sf diameter=2\sqrt{11}\]Ok good good.
So our radius will be half that length.\[\Large\bf\sf radius=\sqrt{11}\]

Answer 17

Then we need to find the location of the center of the sphere.
We can do that using the midpoint formula thing.

Answer 18

Just take the average of each coordinate.

Answer 19

is that (x+y)/2

Answer 20

Sorry x1+x2 / 2^

Answer 21

Yes good, you've got the right idea. Average of each coordinate.
\[\Large\bf\sf midpoint=\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2},\;\frac{z_1+z_2}{2}\right)\]

Answer 22

Thus giving (x-3)+(y-2)+(z-7)=11

Answer 23

Awesome Thanks :D

Answer 24

Yay team \c:/

Answer 25

But the one thing i don<t get is why we need to write = 11 at the end

Answer 26

\[\Large\bf\sf x^2+y^2+z^2=r^2\]We're shifting the center, so that's what the subtraction is about.

See how our radius is being squared?
That's why the square root disappears on the 11.

Answer 27

Ohh I see .. basically it satisfies the general formula

Answer 28

\[\Large\bf\sf r\quad=\quad \sqrt{11}\]\[\Large\bf\sf center\quad=\quad (3,2,7)\]
Yes :) We just needed these two pieces of information so we could put it in standard form.

Answer 29

I probably should have written out the standard form of a sphere with the h, k and l.. but whatever :P

Answer 30

Loll It's perfect I get it :D

Answer 31

Thanks a lot !!!!:D