Graph the ellipse given by the equation
49x2+16y2=784. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.
When an
ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,
(h,k), we use the standard forms
(x−h)2a2+(y−k)2b2=1,a>b for horizontal ellipses and
(x−h)2b2+(y−k)2a2=1,a>b for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.
Given the standard form of an equation for an ellipse centered at
(h,k), sketch the graph.
Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.
If the equation is in the form
(x−h)2a2+(y−k)2b2=1, where
a>b, then
the center is
(h,k)
the major axis is parallel to the
x -axis
the coordinates of the vertices are
(h±a,k)
the coordinates of the co-vertices are
(h,k±b)
the coordinates of the foci are
(h±c,k)
If the equation is in the form
(x−h)2b2+(y−k)2a2=1, where
a>b, then
the center is
(h,k)
the major axis is parallel to the
y -axis
the coordinates of the vertices are
(h,k±a)
the coordinates of the co-vertices are
(h±b,k)
the coordinates of the foci are
(h,k±c)
Solve for
c using the equation
c2=a2−b2.
Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Graphing an ellipse centered at (
h ,
k )
Graph the ellipse given by the equation,
(x+2)24+(y−5)29=1. Identify and label the center, vertices, co-vertices, and foci.
First, we determine the position of the major axis. Because
9>4, the major axis is parallel to the
y -axis. Therefore, the equation is in the form
(x−h)2b2+(y−k)2a2=1, where
b2=4 and
a2=9. It follows that:
the center of the ellipse is
(h,k)=(−2,5)
the coordinates of the vertices are
(h,k±a)=(−2,5±√9)=(−2,5±3), or
(−2,2) and
(−2,8)
the coordinates of the co-vertices are
(h±b,k)=(−2±√4,5)=(−2±2,5), or
(−4,5) and
(0,5)
the coordinates of the foci are
(h,k±c), where
c2=a2−b2. Solving for
c, we have:
c=±√a2−b2=±√9−4=±√5
Therefore, the coordinates of the foci are
(−2,5−√5) and
(−2,5+√5).
Next, we plot and label the center, vertices, co-vertices, and foci, 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.
Recognize that an ellipse described by an equation in the form
ax2+by2+cx+dy+e=0 is in general form.
Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.
Factor out the coefficients of the
x2 and
y2 terms in preparation for completing the square.
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,
m1(x−h)2+m2(y−k)2=m3, where
m1,m2, and
m3 are constants.
Divide both sides of the equation by the constant term to express the equation in standard form.