Lesson Identify elements of a hyperbola given by its general equation
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<H2>Identify elements of a hyperbola given by its general equation</H2> This lesson teaches you by examples on how to identify the center, vertices and foci of a hyperbola given by its general equation. Prerequisites to this lesson are the lessons - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Hyperbola-definition--canonical-equation--characteristic-points-and-elements.lesson>Hyperbola definition, canonical equation, characteristic points and elements</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-a-hyperbola.lesson>Standard equation of a hyperbola</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-a-hyperbola.lesson>General equation of a hyperbola</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-a-hyperbola-to-the-standard-form-by-completing-the-square.lesson>Transform general equation of a hyperbola to the standard form by completing the square</A> in this site. <H3>Problem 1</H3>Find the standard equation, verices and foci of the hyperbola {{{25x^2-39y^2+150x+390y}}} = {{{-225}}}. <B>Solution</B> <pre> {{{25x^2-39y^2+150x+390y}}} = {{{-225}}} ---> (we need to complete squares for x and y separately) ---> {{{(25x^2+150x)-(39y^2-390y)}}} = {{{-225}}} (so we group the x-terms and y-terms separately) {{{25(x^2+6x)-39(y^2-10y)}}} = {{{-225}}} (preparing to complete squares) {{{25(x^2+6x+9)-39(y^2-10y+25)}}} = {{{-225+25*9-39*25}}} (completing the squares) {{{25(x+3)^2-39(y-5)^2}}} = {{{-975}}} (completing the squares is just done) {{{39(y-5)^2-25(x+3)^2}}} = {{{975}}} (finalizing completing the squares) <TABLE> <TR> <TD> {{{39(y-5)^2/975-25(x+3)^2/975}}} = {{{975/975}}} {{{highlight((y-5)^2/25-(x+3)^2/39=1)}}} The standard equation above tells us that the center is at (-3,5) ; the real (or transverse) axis is x= -3 (vertical line parallel to y-axis); the linear eccentricity is {{{c}}} = {{{sqrt(25+39)}}} = {{{sqrt(64)}}} = 8; Since the foci are on the real axis at a distance c = 8 from the center, their coordinates are (-3,5+8) = (-3,1) and (-3,5-8) = (-3,-3). As for the vertices, they also on the real axis, and their coordinates are (-3,5+5) = (-3,10) and (-3,5-5) = (-3,0). </TD> <TD> {{{drawing(360,360,-24,24,-5,15, graph(360,360,-24,24,-5,15, 5+sqrt(1+(x+3)^2/39), 5-sqrt(1+(x+3)^2/39)), circle(-3,5-8,0.2), circle(-3,5,0.2), circle(-3,5+8,0.2), circle(-3,5-8,0.3), circle(-3,5,0.3), circle(-3,5+8,0.3), circle(-3,5-8,0.4), circle(-3,5,0.4), circle(-3,5+8,0.4) )}}} Hyperbola {{{(y-5)^2/25}}} - {{{(x+3)^2/39}}} = 1 </TD> </TR> </TABLE></pre> My lessons on hyperbolas in this site are - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Hyperbola-definition--canonical-equation--characteristic-points-and-elements.lesson>Hyperbola definition, canonical equation, characteristic points and elements</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Hyperbola-focal-property.lesson>Hyperbola focal property</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-and-normal-vectors-to-a-hyperbola.lesson>Tangent lines and normal vectors to a hyperbola</A> - <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-a-hyperbola.lesson>Optical property of a hyperbola</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-a-hyperbola.lesson>Standard equation of a hyperbola</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-a-hyperbola-given-by-its-standard-eqn-NEW.lesson>Identify elements of hyperbola given by its standard equation</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-the-standard-equation-of-a-hyperbola-given-by-its-elements.lesson>Find the standard equation of a hyperbola given by its elements</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-a-hyperbola.lesson>General equation of a hyperbola</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-a-hyperbola-to-the-standard-form-by-completing-the-square.lesson>Transform general equation of a hyperbola to the standard form by completing the square</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/REVIEW-of-lessons-on-hyperbolas.lesson>OVERVIEW of lessons on hyperbolas</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.
