Homework SourseThe surface of the function fxyz zx2y2 that passes through
The surface of the function f(x,y,z) = z-x^2-y^2 that passes through the point (1,2,-3) intersects the (x,z)-plane (y=0) along the curve. Key says the answer is z = x^2 - 8
Solution
The equation for the function using the given point needs to be calculated first :
The function is = z - x^2 - y^2------------(1)
It passes through (1,2,-3), Substituting these values in the function (1) we have :
(-3) - (1^2) - (2^2) = -3 -1 - 4 = -8.
So the equation becomes : z- x^2 - y^2=-8
=> z = (x^2) + (y^2) - 8 ---------- (2)
Now, since it intersects the x-z plane and that is the surface whose equation is needed.
x-z plane => y=0---------------- (3)
Substituting (3) in (2), the surface eqn is :
z = (x^2) + (y^2) - 8
=> z= (x^2) + (0^2) - 8
=> z = x^2 - 8 ------------(ANSWER)
