Paraboloids are generated by rotating a parabola across its axes.
- The equation of the paraboloid is: [tex]\mathbf{z = (x- 1)^2 + (y - 2)^2}[/tex]
- The surface is an infinite paraboloid
The equation is given as:
[tex]\mathbf{x^2 + y^2 - 2x - 4y - z + 5 = 0}[/tex]
Rewrite as:
[tex]\mathbf{x^2 - 2x+ y^2 - 4y - z + 5 = 0}[/tex]
Add 0 to both sides
[tex]\mathbf{x^2 - 2x + 0+ y^2 - 4y - z + 5 = 0 + 0}[/tex]
Express 0 as the difference between the squares of half the coefficient of x
[tex]\mathbf{x^2 - 2x + (\frac{2}{2})^2 - (\frac{2}{2})^2 + y^2 - 4y - z + 5 = 0 + (\frac{2}{2})^2 - (\frac{2}{2})^2}[/tex]
Cancel out the negative term
[tex]\mathbf{x^2 - 2x + (\frac{2}{2})^2 + y^2 - 4y - z + 5 = 0 + (\frac{2}{2})^2 }[/tex]
Simplify
[tex]\mathbf{x^2 - 2x + 1 + y^2 - 4y - z + 5 = 0 +1 }[/tex]
[tex]\mathbf{x^2 - 2x + 1 + y^2 - 4y - z + 5 = 1}[/tex]
Express as squares
[tex]\mathbf{(x- 1)^2 + y^2 - 4y - z + 5 = 1}[/tex]
Repeat the same process for y
[tex]\mathbf{(x- 1)^2 + y^2 - 4y + (2)^2 - z + 5 = 1 + (2)^2}[/tex]
[tex]\mathbf{(x- 1)^2 + y^2 - 4y + 4 - z + 5 = 1 + 4}[/tex]
[tex]\mathbf{(x- 1)^2 + y^2 + (y - 2)^2 - z + 5 = 1 + 4}[/tex]
[tex]\mathbf{(x- 1)^2 + (y - 2)^2 - z + 5 = 1 + 4}[/tex]
[tex]\mathbf{(x- 1)^2 + (y - 2)^2 - z + 5 = 5}[/tex]
Subtract 5 from both sides
[tex]\mathbf{(x- 1)^2 + (y - 2)^2 - z = 0}[/tex]
Make z the subject
[tex]\mathbf{z = (x- 1)^2 + (y - 2)^2}[/tex]
The above equation is the surface of an infinite paraboloid
See attachment for the sketch of the equation.
Read more about paraboloid at:
https://brainly.com/question/14956045
