1. The problem statement, all variables and given/known data

Find the coordinates of the symmetric point of the point M(3,4,7) from the plane 2x-y+z+9=0

2. Relevant equations


3. The attempt at a solution

I found the equation of the plane which the symmetric point is staying at:

2x-y-z+27=0

Also I found the distance between M(3,4,7) and the symmetric point:

\sqrt{(x-3)^2+(y-4)^2+(z-7)^2}=2*3*\sqrt{6}

I need one more condition to solve the system.

The coordinate from my text book results which I need to find is: M_s(-9,10,1)

Your question doesn't make sense, at least to me. Do you mean "Find the reflection image of M in the plane?" or something else? What's 'the symmetric point'?

Yes reflection image of M...

Then you should be able to see that the plane is the collection of points equi-distant from M and it's reflection M'. Therefore, to find M' you find the point on the plane which is nearest to M, P and then you can work out the vector MP. Then M' will be the point P plus MP, since that takes you to the other side of the plane.

I don't quite understand you. Can you define what are the points and where are they lying at?

http://img237.imageshack.us/img237/8596/planetx8.jpg

You know M and you know the equation for the plane. Thus you can find P and therefore M'.

I understand. BUt how I am gonna find P?

Either find the point in the plane closest to M or work out the normal of the plane and find which normal goes through M.

These are methods/processes you should be aware of and should know how to use. Otherwise the question would be impossible for you.

The normal vector is (2,-1,1)
So PM=(2,-1,1)
(3,4,7)-(c,d,f)=(2,-1,1)

(3-c,4-d,7-f)=(2,-1,1)

3-c=2
c=1

4-d=-1
d=5

7-f=1
f=6

So P(1,5,6)

But if I make \sqrt{(x-1)^2+(y-5)^2+(z-6)^2}=\sqrt{6}

and put all in the system, the system got no solution. I don't know what is the problem...

Unless you are very lucky, PM is not equal to the normal but a multiple of the normal. In other words, if n=(2,-1,-1) is the normal, then PM=kn with some real number k. Now find k from the condition that P should lie on the plane.

Hm... What is that condition?

I found the point (0, \frac{11}{2}, \frac{11}{2}) and again the system got no solution...