2.6 Quadric surfaces  (Page 5/15)

$\text{−}{x}^2+36y^2+36z^2=9$

$\mathrm{-4}{x}^2+25y^2+z^2=100$

$\mathrm{-3}{x}^2+5y^2-z^2=10$

$3x^2-y^2-6z^2=18$

$5y=x^2-z^2$

$8x^2-5y^2-10z=0$

$x^2+5y^2+3z^2-15=0$

$63x^2+7y^2+9z^2-63=0$

$x^2+5y^2-8z^2=0$

$5x^2-4y^2+20z^2=0$

$6x=3y^2+2z^2$

$49y=x^2+7z^2$

[T] $x^2+z^2+4y=0,z=0$

[T] $x^2+z^2+4y=0,x=0$

[T] $\mathrm{-4}{x}^2+25y^2+z^2=100,x=0$

[T] $\mathrm{-4}{x}^2+25y^2+z^2=100,y=0$

[T] $x^2+\frac{y^2}{4}+\frac{z^2}{100}=1,x=0$

[T] $x^2-y-z^2=1,y=0$

Use the graph of the given quadric surface to answer the questions.

1. Specify the name of the quadric surface.
2. Which of the equations— $16x^2+9y^2+36z^2=3600,9x^2+36y^2+16z^2=3600,$ or $36x^2+9y^2+16z^2=3600$ —corresponds to the graph?
3. Use b. to write the equation of the quadric surface in standard form.

Use the graph of the given quadric surface to answer the questions.

1. Specify the name of the quadric surface.
2. Which of the equations— $36z=9x^2+y^2,9x^2+4y^2=36z,\text{or}-36z=\mathrm{-81}{x}^2+4y^2$ —corresponds to the graph above?
3. Use b. to write the equation of the quadric surface in standard form.

$x^2+2z^2+6x-8z+1=0$

$4x^2-y^2+z^2-8x+2y+2z+3=0$

$x^2+4y^2-4z^2-6x-16y-16z+5=0$

$x^2+z^2-4y+4=0$

$x^2+\frac{y^2}{4}-\frac{z^2}{3}+6x+9=0$

$x^2-y^2+z^2-12z+2x+37=0$

Write the standard form of the equation of the ellipsoid centered at the origin that passes through points $A\left(2,0,0\right),B\left(0,0,1\right),$ and $C\left(\frac{1}{2},\sqrt{11},\frac{1}{2}\right).$

Write the standard form of the equation of the ellipsoid centered at point $P\left(1,1,0\right)$ that passes through points $A\left(6,1,0\right),B\left(4,2,0\right)$ and $C\left(1,2,1\right).$

Determine the intersection points of elliptic cone $x^2-y^2-z^2=0$ with the line of symmetric equations $\frac{x-1}{2}=\frac{y+1}{3}=z.$

Determine the intersection points of parabolic hyperboloid $z=3x^2-2y^2$ with the line of parametric equations $x=3t,y=2t,z=19t,$ where $t\in ℝ.$

Find the equation of the quadric surface with points $P\left(x,y,z\right)$ that are equidistant from point $Q\left(0,\mathrm{-1},0\right)$ and plane of equation $y=1.$ Identify the surface.

Find the equation of the quadric surface with points $P\left(x,y,z\right)$ that are equidistant from point $Q\left(0,2,0\right)$ and plane of equation $y=\mathrm{-2}.$ Identify the surface.

If the surface of a parabolic reflector is described by equation $400z=x^2+y^2,$ find the focal point of the reflector.