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Answer:
Required equation of tangent plane is
.
Step-by-step explanation:
Given surface function is,
To find tangent plane at the point (5,-1,1).
We know equation of tangent plane at the point $(x_0,y_0,z_0)[/tex] is,
![z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)\hfill (1)](https://tex.z-dn.net/?f=z%3Df%28x_0%2Cy_0%29%2Bf_x%28x_0%2Cy_0%29%28x-x_0%29%2Bf_y%28x_0%2Cy_0%29%28y-y_0%29%5Chfill%20%281%29)
So that,
![f(x_0,y_0)=6-\frac{6}{5}(5+1)=-\frac{6}{5}](https://tex.z-dn.net/?f=f%28x_0%2Cy_0%29%3D6-%5Cfrac%7B6%7D%7B5%7D%285%2B1%29%3D-%5Cfrac%7B6%7D%7B5%7D)
![f_x=-\frac{6}{5}y\implies f_x(5,-1,1)=\frac{6}{5}](https://tex.z-dn.net/?f=f_x%3D-%5Cfrac%7B6%7D%7B5%7Dy%5Cimplies%20f_x%285%2C-1%2C1%29%3D%5Cfrac%7B6%7D%7B5%7D)
![f_y=-\frac{6}{5}x\implies f_y(5,-1,1)=-6](https://tex.z-dn.net/?f=f_y%3D-%5Cfrac%7B6%7D%7B5%7Dx%5Cimplies%20f_y%285%2C-1%2C1%29%3D-6)
Substitute all these values in (1) we get,
![z=\frac{6}{5}(x-5)-6(y+1)-\frac{6}{5}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B6%7D%7B5%7D%28x-5%29-6%28y%2B1%29-%5Cfrac%7B6%7D%7B5%7D)
![\therefore z=\frac{6}{5}(x-5y-11)](https://tex.z-dn.net/?f=%5Ctherefore%20z%3D%5Cfrac%7B6%7D%7B5%7D%28x-5y-11%29)
Which is the required euation of tangent plane.
Assuming that CB is tangent, this is just a right triangle with a hypotenuse of 20 and a side of 12 so by the Pythagorean Theorem:
20^2=12^2+x^2
400=144+x^2
x^2=400-144
x^2=256
x=16