12.1 The ellipse  (Page 6/16)

Given the standard form of an equation for an ellipse centered at $\left(h,k\right),$ sketch the graph.

1. Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.
   1. If the equation is in the form $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1,$ where $a>b,$ then
      • the center is $\left(h,k\right)$
      • the major axis is parallel to the x -axis
      • the coordinates of the vertices are $\left(h\pm a,k\right)$
      • the coordinates of the co-vertices are $\left(h,k\pm b\right)$
      • the coordinates of the foci are $\left(h\pm c,k\right)$
   2. If the equation is in the form $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1,$ where $a>b,$ then
      • the center is $\left(h,k\right)$
      • the major axis is parallel to the y -axis
      • the coordinates of the vertices are $\left(h,k\pm a\right)$
      • the coordinates of the co-vertices are $\left(h\pm b,k\right)$
      • the coordinates of the foci are $\left(h,k\pm c\right)$
2. Solve for $c$ using the equation $c^2=a^2-b^2.$
3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.

Given the general form of an equation for an ellipse centered at ( h , k ), express the equation in standard form.

1. Recognize that an ellipse described by an equation in the form $a{x}^2+b{y}^2+cx+dy+e=0$ is in general form.
2. Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.
3. Factor out the coefficients of the $x^2$ and $y^2$ terms in preparation for completing the square.
4. Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant, $m_1{\left(x-h\right)}^2+m_2{\left(y-k\right)}^2=m_3,$ where $m_1,m_2,$ and $m_3$ are constants.
5. Divide both sides of the equation by the constant term to express the equation in standard form.