Find the equation for the tangent plane to the surface x^2yz+3y^2=2xz^2-8z at the point (1,2,-1)?

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freemihai | College Teacher | (Level 2) Adjunct Educator

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I will start with the final form of equation you must arrive at. Take a look at this equation, get used to it and then we will understand together what we must do to reach to this form:

`N(r - r_0) = 0`

If you are confused, do not worry at all. You will understand right away what is the meaning of this equation. As you can see, N and `(r - r_0)`  represent two vectors, N is the normal vector to the plane `x^2yz+3y^2=2xz^2-8z`  and `r - r_0`  is the vector that lies in the tangent plane. Therefore, the product `N(r - r_0) = 0`  must be the dot product of the vectors.

Why the dot product is equal to 0? Well, the answer is simple. The dot product is 0 because the normal to the surface is normal to the tangent plane, too.

All you need to do is to start looking for the vector equation of normal N.

N = `grad` `F|_(1,2,-1) = F_x|_(1,2,-1) + F_y|_(1,2,-1) + F_z|_(1,2,-1)`

`N = (2xyz - 2z^2)|_(1,2,-1)*i + (x^2z+6y)|_(1,2,-1)*j + (xy-4xz+8)|_(1,2,-1)*k`

Just put 1,2,-1 instead of x,y,z and start calculate partial derivatives:

`N = -6i+11j+14k`

Now, it is the time to calculate the equations of vectors `r_0`  and r:

`r_0 = i+2j-k`

`r = x i + yj + zk`

Calculate `r-r_0`  subtracting the coefficients of i,j,k;

`r-r_0 = (x-1)i+(y-2)j+(z+1)k`

Now it's time for the dot product:

`N*(r-r_0) = 0`

Multiplicate coefficients of i,j,k of vectors N and `r-r_0` :

-`6(x-1) + 11(y-2) + 14(z+1 ) = 0`


The answer to this problem: `6x-11y-14z+2=0`

If you have any questions regarding the explanation, please feel free to ask what you did not quite understood.