Question

is Find an equation for, and identify, the surface that results when the surface x2 - y2 – 2z2 reflected about the plane Y = 2. b) Find an equation for, and identify, the surface that results when the surface x2 – 4y2 + x2 = ( is translated to the point D(2,-1,-4).

Answer 1

The given equation can be rewritten as :

$\frac{y^2}{2}+ z^2-\frac{x^2}{2}=1$

which is the equation of a hyperboloid of one sheet along the x- axis, like one in the picture:

Z у Х

Reflecting this surface about the plane $y=z$ will result in the same surface , since the surface is symmetric about the plane . Hence the resulting equation will be

or

یا 12 - 2 - 22 = -2

$\frac{y^2}{2}+ z^2-\frac{x^2}{2} =1$

(b) The given equation can be rewritten as:

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

which is the equation of an elliptic cone with axis y-axis. Translating the origin to D, we get the equation:

$\frac{(x-2)^2}{4}+\frac{(z+4)^2}{4}-(y-1)^2=0$

which is the equation of an elliptic cone with centre D