Translation of axes

Given the equation

${\displaystyle 9x^{2}+25y^{2}+18x-100y-116=0,}$

by using a translation of axes, determine whether the locus of the equation is a parabola, ellipse, or hyperbola. Determine foci (or focus), vertices (or vertex), and eccentricity.

Solution: To complete the square in x and y, write the equation in the form

${\displaystyle 9(x^{2}+2x\qquad )+25(y^{2}-4y\qquad )=116.}$

Complete the squares and obtain

${\displaystyle 9(x^{2}+2x+1)+25(y^{2}-4y+4)=116+9+100}$

${\displaystyle \Leftrightarrow 9(x+1)^{2}+25(y-2)^{2}=225.}$

Define

${\displaystyle x'=x+1}$      and      ${\displaystyle y'=y-2.}$

That is, the translation in equations (2) is made with ${\displaystyle h=-1,k=2.}$ The equation in the new coordinate system is

${\displaystyle 9x'^{2}+25y'^{2}=225.}$

(4)

Divide equation (4) by 225 to obtain

${\displaystyle {\frac {x'^{2}}{25}}+{\frac {y'^{2}}{9}}=1,}$

which is recognizable as an ellipse with ${\displaystyle a=5,b=3,c^{2}=a^{2}-b^{2}=16,c=4,e={\tfrac {4}{5}}.}$ In the x'y'-system, we have: center ${\displaystyle (0,0)}$; vertices ${\displaystyle (\pm 5,0)}$; foci ${\displaystyle (\pm 4,0).}$

In the xy-system, use the relations ${\displaystyle x=x'-1,y=y'+2}$ to obtain: center ${\displaystyle (-1,2)}$; vertices ${\displaystyle (4,2),(-6,2)}$; foci ${\displaystyle (3,2),(-5,2)}$; eccentricity ${\displaystyle {\tfrac {4}{5}}.}$[8]

Use a translation of coordinates to identify the quadric surface

${\displaystyle x^{2}+4y^{2}+3z^{2}+2x-8y+9z=10.}$

Solution: Write the equation in the form

${\displaystyle x^{2}+2x\qquad +4(y^{2}-2y\qquad )+3(z^{2}+3z\qquad )=10.}$

Complete the square to obtain

${\displaystyle (x+1)^{2}+4(y-1)^{2}+3(z+{\tfrac {3}{2}})^{2}=10+1+4+{\tfrac {27}{4}}.}$

Introduce the translation of coordinates

${\displaystyle x'=x+1,\qquad y'=y-1,\qquad z'=z+{\tfrac {3}{2}}.}$

The equation of the surface takes the form

${\displaystyle x'^{2}+4y'^{2}+3z'^{2}={\tfrac {87}{4}},}$

which is recognizable as the equation of an ellipsoid.[13]