I tied to answer the following questions is about the curve surface $z= f (x, y) = x^2 + y^2$ in the xyz space. And the three questions related to each other
A.) Find the tangent plane equation at the point $(a, b, a^2+ b^2) $in curved surface z . The equation of the tangent plane at the point $(a, b, f (a, b)) $ on z is given by the following equation
$Z-f(a,b)=f_x(a,b)(x-a)+f_y(a,b)(x-b)$
So i got
$Z-(a^2+b^2)=2a(x-a)+2b(x-b)$ $2ax+2by-(a^2+b^2)$
2) when the tangent plane of the previous question moves pass through the point (0,0,-1). Find The equation for a plane S that contains the contact trajectory.
Tried to put 0,0,-1 to equation in number 1 $-1=-(a^2+b^2)$ $z=1$ is S(?)
But i wasnt so sure what is S plane here and what is the relation with Z?
- Calculate the volume V of the part surrounded by $z=x^2+y^2 $ and the plane S
Note : I was confuse about number 3, what is the area surrounded by S and Z (?) Since i wasnt so sure about S and Z here