Lesson Identify elements of an ellipse given by its standard equation
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<H2>Identify elements of an ellipse given by its standard equation</H2> This lesson teaches you by examples on how to identify axes, semi-axes, vertices, co-vertices and foci of the ellipse given by its standard equation. Your prerequisite is this lesson <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-definition--canonical-equation--characteristic-points-and-elements.lesson>Ellipse definition, canonical equation, characteristic points and elements</A>. <pre> An equation of the form {{{(x-h)^2/a^2}}} + {{{(y-k)^2/b^2}}} = 1, a > b > 0 is the standard equation of an ellipse with the center at the point (h,k) Under the condition a > b > 0, this ellipse has the major axis parallel to x-axis and the minor axis parallel to y-axis. At this condition, the major semi-axis has the length "a" and the minor semi-axis has the length "b". The ellipse is wider than tall. The linear eccentricity is c = {{{sqrt(a^2 - b^2)}}}. The foci are located at (h-c,k) and (h+c,k). The vertices are located at (h-a,k) and (h+a,k). The co-vertices are located at (h,k+b) and (h,k-b). An equation of the form {{{(x-h)^2/a^2}}} + {{{(y-k)^2/b^2}}} = 1, b > a > 0 is the standard equation of an ellipse with the center at the point (h,k), too. Under the condition b > a > 0, this ellipse has the major axis parallel to y-axis and the minor axis parallel to x-axis. At this condition, the major semi-axis has the length "b" and the minor semi-axis has the length "a". The ellipse is taller than wide. The linear eccentricity is c = {{{sqrt(b^2 - a^2)}}}. The foci are located at (h,k-c) and (h,k+c), The vertices are located at (h,k-b) and (h,k+b). The co-vertices are located at (h+a,k) and (h-a,k). </pre> <H3>Problem 1</H3>Identify the center, axes , semi-axes, vertices, co-vertices, and foci of the ellipse given by its standard equation {{{(x-2)^2/9}}} + {{{y^2/4}}} = {{{1}}}. <B>Solution</B> <pre> The center is at the point (2,0). The major axis lies on x-axis, while the minor axis is parallel to y-axis. The major semi-axis has the length of a= 3 units. The minor semi-axis has the length of b= 2 units. The vertices are (5,0) and (-1,0), while the co-vertices are (2,2) and (2,-2). The linear eccentricity is {{{c}}} = {{{sqrt(a^2 - b^2)}}} = {{{sqrt(3^2 - 2^2)}}} = {{{sqrt(9-4)}}} = {{{sqrt(5)}}}. The foci of the ellipse are ({{{2-sqrt(5)}}},{{{0}}}) and ({{{2+sqrt(5)}}},{{{0}}}). </pre><TABLE> <TR> <TD> {{{graph( 330, 270, -3.5, 8.5, -4.5, 4.5, 2*sqrt(1-((x-2)^2/9)), -2*sqrt(1-((x-2)^2/9)) )}}} Ellipse {{{(x-2)^2/9}}} + {{{y^2/4}}} = 1 </TD> </TR> </TABLE> <H3>Problem 2</H3>Identify the center, axes, semi-axes, vertices, co-vertices, and foci of the ellipse given by its standard equation {{{x^2/9}}} + {{{(y+1)^2/16}}} = {{{1}}}. <B>Solution</B> <pre> The center is at the point (0,-1). The major axis lies on y-axis, while the minor axis is parallel to x-axis. The major semi-axis has the length of a= 4 units. The minor semi-axis has the length of b= 3 units. The vertices are (0,3) and (0,-5), while the co-vertices are (-3,-1) and (3,-1). The linear eccentricity is {{{c}}} = {{{sqrt(b^2 - a^2)}}} = {{{sqrt(4^2 - 3^2)}}} = {{{sqrt(16-9)}}} = {{{sqrt(7)}}}. The foci of the ellipse are ({{{0}}},{{{-1-sqrt(7)}}}) and ({{{0}}},{{{-1+sqrt(7)}}}). </pre><TABLE> <TR> <TD> {{{graph( 330, 270, -5.5, 5.5, -5.5, 3.5, -1 + 4*sqrt(1-(x^2/9)), -1 - 4*sqrt(1-(x^2/9)) )}}} Ellipse {{{x^2/9}}} + {{{(y+1)^2/16}}} = 1 </TD> </TR> </TABLE> <H3>Problem 3</H3>Identify the center, axes , semi-axes, vertices, co-vertices, and foci of the ellipse given by its standard equation {{{(x-3)^2/9}}} + {{{(y-1)^2/4}}} = {{{1}}}. <B>Solution</B> <pre> The center is at (3,1) in the coordinate plane. The major axis is parallel to x-axis, while the minor axis is parallel to y-axis. The major semi-axis has the length of a= 3 units. The minor semi-axis has the length of b= 2 units. The vertices are (0,1) and (6,1). The co-vertices are (3,3) and (3,-1). The linear eccentricity is {{{c}}} = {{{sqrt(a^2 - b^2)}}} = {{{sqrt(3^2 - 2^2)}}} = {{{sqrt(9-4)}}} = {{{sqrt(5)}}}. The foci of the ellipse are at ({{{3-sqrt(5)}}},{{{1}}}) and ({{{3+sqrt(5)}}},{{{1}}}). </pre><TABLE> <TR> <TD> {{{graph( 330, 270, -2.5, 8.5, -3.5, 5.5, 1 + 2*sqrt(1-((x-3)^2/9)), 1 -2*sqrt(1-(x-3)^2/9)) )}}} Ellipse {{{(x-3)^2/9}}} + {{{(y-1)^2/4}}} = 1 </TD> </TR> </TABLE> <H3>Problem 4</H3>Identify the center, axes, semi-axes, vertices, co-vertices, and foci of the ellipse given by its standard equation {{{(x+2)^2/9}}} + {{{(y+1)^2/16}}} = {{{1}}}. <B>Solution</B> <pre> The center is at the point (-2,-1) of the coordinate plane. The major axis parallel to y-axis, while the minor axis is parallel to x-axis. The major semi-axis has the length of a= 4 units. The minor semi-axis has the length of b= 3 units. The vertices are (-2,-5) and (-2,3), while the co-vertices are (-5,-1) and (1,-1). The linear eccentricity is {{{c}}} = {{{sqrt(b^2 - a^2)}}} = {{{sqrt(4^2 - 3^2)}}} = {{{sqrt(16-9)}}} = {{{sqrt(7)}}}. The foci of the ellipse are ({{{-2}}},{{{-1-sqrt(7)}}}) and ({{{-2}}},{{{-1+sqrt(7)}}}). </pre><TABLE> <TR> <TD> {{{graph( 330, 300, -7.5, 3.5, -5.5, 4.5, -1+4*sqrt(1-((x+2)^2/9)), -1-4*sqrt(1-((x+2)^2/9)) )}}} Ellipse {{{(x+3)^2/9}}} + {{{(y+1)^2/16}}} = 1 </TD> </TR> </TABLE> My other lessons on ellipses in this site are - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-definition--canonical-equation--characteristic-points-and-elements.lesson>Ellipse definition, canonical equation, characteristic points and elements</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-focal-property.lesson>Ellipse focal property</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangen-lines-to-a-circle.lesson>Tangent lines and normal vectors to a circle</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-to-an-ellipse.lesson>Tangent lines and normal vectors to an ellipse</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-an-ellipse.lesson>Optical property of an ellipse</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-an-ellipse-revisited.lesson>Optical property of an ellipse revisited</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-an-ellipse.lesson>Standard equation of an ellipse</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-a-standard-equation-of-an-ellipse-given-by-its-elements.lesson>Find the standard equation of an ellipse given by its elements</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-an-ellipse.lesson>General equation of an ellipse</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-an-ellipse-to-the-standard-form-by-completing-the-square.lesson>Transform a general equation of an ellipse to the standard form by completing the square</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-vertices-co-vertices-foci-of-the-ellipse-given-by-an-equation.lesson>Identify elements of an ellipse given by its general equation</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-a-circle.lesson>Standard equation of a circle</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-the-equation-of-the-circle-given-by-its-center-and-tauching-a-given-line.lesson>Find the standard equation of a circle</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-a-circle.lesson>General equation of a circle</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-a-circle-to-the-standard-form-by-completing-the-squares.lesson>Transform general equation of a circle to the standard form by completing the squares</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-a-circle-given-by-its-general-equation.lesson>Identify elements of a circle given by its general equation</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/REVIEW-of-lessons-on-ellipses.lesson>OVERVIEW of lessons on ellipses</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
