Respuesta :
Answer:
(1,-1,2) lies on the surface.
Therefore a vector to the normal to the surface is
[tex]\hat{n}=2\hat{i}+3\hat{j}+2\hat{k}[/tex]
Therefore the equation of tangent plane is
2x+3y+2z=3
Step-by-step explanation:
Given equation of surface is
2x²-3y²+z²=3
Given point is (1,-1,2)
To check that whether the point lies on the surface or not .
We have to put x=1 , y= -1 and z= 2 in the given surface.
L.H.S
2(1)²-3(-1)²+2²
=2-3+4
=3 = R.H.S
Since the point (1,-1,2) point satisfies the equation.
Therefore (1,-1,2) lies on the surface.
Here f(x,y,z)= 2x²-3y²+z²
To find the a vector normal to the surface we have to find
[tex]f_x=\frac{\partial f }{\partial x}[/tex] [ where only variable is x]
[tex]f_y=\frac{\partial f }{\partial y}[/tex] [ where only variable y]
[tex]f_z=\frac{\partial f }{\partial z}[/tex] [ where only variable z]
[tex]f_x=\frac{\partial f }{\partial x}=\frac{\partial }{\partial x}(2x^2-3y^2+z^2)[/tex] = 4x
[tex]f_y=\frac{\partial f }{\partial y}=\frac{\partial }{\partial y}(2x^2-3y^2+z^2)=-6y[/tex]
[tex]f_z=\frac{\partial f }{\partial z}=\frac{\partial }{\partial z}(2x^2-3y^2+z^2)=2z[/tex]
The gradient at (1,-1,2)
[tex]\bigtriangledown f(1,-1,2)\\= (4\times 1)\hat{i} +[-6\times (-1)]\hat{j}+(2\times 2)\hat{k}[/tex] [ putting x=1,y=-1 and z=2 in
[tex]=4\hat{i}+6\hat{j}+4\hat{k}[/tex] [tex]f_x,f_y \ and \ f_z[/tex]]
Therefore a vector to the normal to the surface is
[tex]\hat{n}=4\hat{i}+6\hat{j}+4\hat{k}[/tex]
[tex]or,\hat{n}=2\hat{i}+3\hat{j}+2\hat{k}[/tex] [ remove the common part= 2]
The equation of tangent plane is
[tex]\vec{r}.\hat{n}=\vec {a}.\hat{n}[/tex]
[tex]\vec{r}= x\hat{i}+y\hat{j}+z\hat{k}[/tex]
[tex]\hat{n}[/tex] = normal vector
[tex]\vec{a}[/tex] = the position vector of the given point
Here [tex]\hat{n}=2\hat{i}+3\hat{j}+2\hat{k}[/tex] and [tex]\vec a= \hat i-\hat j+2\hat k[/tex]
Therefore the equation of tangent plane is
[tex]( x\hat{i}+y\hat{j}+z\hat{k}).(2\hat{i}+3\hat{j}+2\hat{k})=( \hat i-\hat j+2\hat k). (2\hat{i}+3\hat{j}+2\hat{k})[/tex]
⇒2.x+3.y+2.z=(1.2)+(-1)(3)+2.2
⇒2x+3y+2z=2-3+4
⇒2x+3y+2z=3
