Answer
Volume of the pyramids cut off from the first octant by any tangent planes to the surface $xyz=1$ at points in the first octant is same.
Work Step by Step
Our aim is to determine the tangent plane equation.
The general form is: $(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$ ...(1)
From the given data, we have $f(x,y,z)=(a,b,c)$
Equation (1), becomes:
Thus,
$(x-a)bc)+(y-b)ac+(z-c)ab=0$
or, $3abc=bcx+acy+abz$
Next step is to determine x-,y-,z- intercepts.Thus,
x-intercept: 3a; y-intercept: 3b and z-intercept: 3c
Volume of the pyramid,$V=xyz=abc=\dfrac{1}{3}(\dfrac{9ab}{2})(3c)=\dfrac{9abc}{2}$
This yields $V=\dfrac{9}{2}$
Hence, the result.The volume of the pyramids cut off from the first octant by any tangent planes to the surface $xyz=1$ at points in the first octant is same.
