sample 8 : Equation of Ellipse

College algebra problems on the equation of the ellipse are presented. More Problems on ellipses with detailed solutions are included in this site. The solutions to the questions below are at the bottom of the page.

a) the major axis and the minor axis of the ellipse and their lengths,

b) the vertices of the ellipse,

c) and the foci of this ellipse.

b) Find the equation of part of the graph of the given ellipse that is below the x axis.

a) its center,

a) its major and minor axes and their lengths,

b) its vertices,

c) and the foci.

a) the center of the ellipse,

b) its major and minor axes and their lengths,

c) its vertices,

d) and its foci.

with \( a = \dfrac{3}{5} \) and \( b = \dfrac{1}{3} \). The major axis is the x axis and its length is equal to \( 2a = \dfrac{6}{5} = 1.2 \)

b) The vertices are on the major axis at the points \( (0 , a) = (0 , 4) \) and \( (0 , -a) = (0 , -4) \)

c) The foci are on the major axis at the points \( (0 , c) \) and \( (0 , -c) \) such that

\( c^{2} = a^{2} - b^{2} = 12 \).

Hence the foci are at the points \( (0 , 2\sqrt{3}) \) and \( (0 , -2\sqrt{3}) \)

\( b^{2} = a^{2} - c^{2} = 36 - 16 = 20 \)

\( b = 2\sqrt{5} \)

The equation of the ellipse is given by \[ \dfrac{x^{2}}{20} + \dfrac{y^{2}}{36} = 1 \]

The equation of the ellipse is given by \[ \dfrac{x^{2}}{24} + \dfrac{y^{2}}{49} = 1 \]

\( x = \sqrt{1 - 3y^{2}} \)

b) Solve the ellipse equation for \( y \) and select the solution for which \( y \) is negative.

\( y = - \sqrt{\dfrac{1}{3} - \dfrac{1}{3}x^{2}} \)

a) its major axis is the line \( x = 1 \), and its minor is the line \( y = -4 \). length of major axis = \( 2a = 8 \) , length of minor axis = \( 2b = 6 \)

b) vertices at: \( (1 , -4 + 4) = (1 , 0) \) and \( (1 , -4 - 4) = (1 , -8) \)

c) \( c = \sqrt{a^{2} - b^{2}} = \sqrt{7} \)

Foci at: \( (1 , -4 + \sqrt{7}) \) and \( (1 , -4 - \sqrt{7}) \)

The center of the ellipse is the midpoint of the two foci and is at \( (2 , 0) \).

\( c \) is the length from one foci to the center, hence \( c = 2 \).

length of minor axis 2 = \( 2b \) hence \( b = 1 \)

\( a^{2} = b^{2} + c^{2} = 5 \)

Since the foci are on the x axis, the major axis of the ellipse is the x axis.

Equation of ellipse:

\( x / 6 = \sin(t) \) and \( y / 4 = \cos(t) \)

Square both sides of the two equations: \( (x / 6)^{2} = \sin^{2}(t) \) and \( (y / 4)^{2} = \cos^{2}(t) \)

Use the fact that \( \sin^{2}(t) + \cos^{2}(t) = 1 \) to write

\( x^{2} / 36 + y^{2} / 16 = 1 \)

\( a = 6 \), \( b = 4 \), \( c = \sqrt{36 - 16} = 2\sqrt{5} \)

major axis: x axis , length = 12

minor axis: y axis , length = 8

Problems on ellipses

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