document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=32163>Question 32163</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question:\r<BR>\n" );
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document.write( "How do I find the vertex and intercepts for each parabola.<BR>\n" );
document.write( "g(x)=x^2+x-6\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #18690 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=18690\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> SEE THE FOLLOWING AND TRY.IF STILL IN DIFFICULTY PLEASE COME BACK<BR>\n" );
document.write( "Linear_Algebra/30362: Question: Find the equation of the ellipse whose center is (5,-3) that has a vertex at 13,-3) and a minor axis of lenght 10.						<BR>\n" );
document.write( "POssible Answers:						<BR>\n" );
document.write( "(A) (x-5)^2/64 + (y+3)^2/25 = 1						<BR>\n" );
document.write( "(B) (x+5)^2/64 + (y-3)^2/25 = 1						<BR>\n" );
document.write( "(C) x^2/64 + y^2/25 = 1						<BR>\n" );
document.write( "(D) none of these						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 17014 by venugopalramana(1167) About Me  on 2006-03-15 11:21:03 (Show Source):						<BR>\n" );
document.write( "SEE THE FOLLOWING AND TRY..IF STILL IN DIFFICULTY PLEASE COME BACK...						<BR>\n" );
document.write( "OK I WORKED IT OUT FOR YOU NOW						<BR>\n" );
document.write( "I TOLD YOU EQN IS						<BR>\n" );
document.write( "(X-H)^2/A^2 + (Y-K)^2/B^2....						<BR>\n" );
document.write( "WHERE H,K IS CENTRE...SO H=5 AND K=-3 AS CENTRE IS GIVEN AS (5,-3)....NOW VERTEX IS (13,-3)...IT LIES ON ELLIPSE..SO IT SATISFIES THE EQN						<BR>\n" );
document.write( "(13-5)^2/A^2 +(-3+3)^2/B^2 =1						<BR>\n" );
document.write( "HENCE A^2=64...OR A=8						<BR>\n" );
document.write( "MINOR AXIS =10=2B...HENCE B=5..SO EQN.S						<BR>\n" );
document.write( "(X-H)^2/64 + (Y+3)^2/25 =1						<BR>\n" );
document.write( "THAT IS A IS CORRECT.						<BR>\n" );
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document.write( "Can you help me write an equation for an ellipse with a major axis with endpoints of (0,8), and (0,-8) with foci of (0,5) and (0,-5)?						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
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document.write( "Answer 16810 by venugopalramana(1120) on 2006-03-13 11:19:12 (Show Source):						<BR>\n" );
document.write( "Can you help me write an equation for an ellipse with a major axis with endpoints of (0,8), and (0,-8) with foci of (0,5) and (0,-5)?						<BR>\n" );
document.write( "THIS SHOWS THAT X AXIS IS THE MAJOR AXIS						<BR>\n" );
document.write( "STANDARD EQN.OF ELLIPSE IS						<BR>\n" );
document.write( "(X-H)^2/A^2 +(Y-K)^2/B^2=1						<BR>\n" );
document.write( "CENTRE IS (H,K)..AS PER THE PROBLEM H=K=0 AS CENTRE OF ELLIPSE IS AT (0,0)..SINCE major axis with endpoints ARE (0,8), and (0,-8)						<BR>\n" );
document.write( "WHERE MAJOR AXIS =2A=8+8=16...SO A=8..SINCE major axis with endpoints ARE (0,8), and (0,-8)						<BR>\n" );
document.write( "FOCI ARE GIVEN BY						<BR>\n" );
document.write( "AE,0 AND -AE,0...SO AE =5...SO E=5/A=5/8						<BR>\n" );
document.write( "BUT E=SQRT{(A^2-B^2)/A^2}=5/8...SQUARING						<BR>\n" );
document.write( "25/64=(A^2-B^2)/A^2=1-B^2/A^2						<BR>\n" );
document.write( "B^2/64=1-25/64=49/64						<BR>\n" );
document.write( "B^2=49						<BR>\n" );
document.write( "B=7						<BR>\n" );
document.write( "HENCE EQN. OF ELLIPSE IS						<BR>\n" );
document.write( "X^2/64 + Y^2/49 = 1						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/30009: Can you help me write an equation for an ellipse with a major axis with endpoints of (0,8), and (0,-8) with foci of (0,5) and (0,-5)?						<BR>\n" );
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document.write( "1 solutions						<BR>\n" );
document.write( "Answer 16810 by venugopalramana(1167) About Me  on 2006-03-13 11:19:12 (Show Source):						<BR>\n" );
document.write( "Can you help me write an equation for an ellipse with a major axis with endpoints of (0,8), and (0,-8) with foci of (0,5) and (0,-5)?						<BR>\n" );
document.write( "THIS SHOWS THAT X AXIS IS THE MAJOR AXIS						<BR>\n" );
document.write( "STANDARD EQN.OF ELLIPSE IS						<BR>\n" );
document.write( "(X-H)^2/A^2 +(Y-K)^2/B^2=1						<BR>\n" );
document.write( "CENTRE IS (H,K)..AS PER THE PROBLEM H=K=0 AS CENTRE OF ELLIPSE IS AT (0,0)..SINCE major axis with endpoints ARE (0,8), and (0,-8)						<BR>\n" );
document.write( "WHERE MAJOR AXIS =2A=8+8=16...SO A=8..SINCE major axis with endpoints ARE (0,8), and (0,-8)						<BR>\n" );
document.write( "FOCI ARE GIVEN BY						<BR>\n" );
document.write( "AE,0 AND -AE,0...SO AE =5...SO E=5/A=5/8						<BR>\n" );
document.write( "BUT E=SQRT{(A^2-B^2)/A^2}=5/8...SQUARING						<BR>\n" );
document.write( "25/64=(A^2-B^2)/A^2=1-B^2/A^2						<BR>\n" );
document.write( "B^2/64=1-25/64=49/64						<BR>\n" );
document.write( "B^2=49						<BR>\n" );
document.write( "B=7						<BR>\n" );
document.write( "HENCE EQN. OF ELLIPSE IS						<BR>\n" );
document.write( "X^2/64 + Y^2/49 = 1						<BR>\n" );
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document.write( "Equations/30056: I need to determine the following for these two problems :HOW MANY X-INTERCEPTS THE PARABOLA HAS, and WHETER ITS VERTEX LIES ABOVE OR BELOW OR ON THE X-AXIS.						<BR>\n" );
document.write( "1.problem						<BR>\n" );
document.write( "y=x^2-5x+6						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "2.problem						<BR>\n" );
document.write( "y=-x^2+2x-1						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 16805 by venugopalramana(1167) About Me  on 2006-03-13 10:41:52 (Show Source):						<BR>\n" );
document.write( "I need to determine the following for these two problems :HOW MANY X-INTERCEPTS THE PARABOLA HAS, and WHETER ITS VERTEX LIES ABOVE OR BELOW OR ON THE X-AXIS.						<BR>\n" );
document.write( "1.problem						<BR>\n" );
document.write( "y=x^2-5x+6						<BR>\n" );
document.write( "PUT Y=0 AND SOLVE FOR X TO GET X INTERCEPTS.						<BR>\n" );
document.write( "X^2-5X+6=0=X^2-2X-3X+6=0=X(X-2)-3(X-2)=0=(X-2)(X-3)=0....X=2 AND 3...						<BR>\n" );
document.write( "SO THE X INTERCEPTS ARE AT X = 2 AND X = 3						<BR>\n" );
document.write( "Y=X^2-5X+6={X^2-2*X*5/2+(5/2)^2}-(5/2)^2+6 =(X-2.5)^2 - 0.25						<BR>\n" );
document.write( "SO THE VERTEX IS AT X=2.5 AND Y=-0.25...THAT IS BELOW X AXIS						<BR>\n" );
document.write( "2.problem						<BR>\n" );
document.write( "y=-x^2+2x-1						<BR>\n" );
document.write( "DOING THE SAME WAY WE GET						<BR>\n" );
document.write( "Y=-(X-1)^2=0 AND HENCE						<BR>\n" );
document.write( "X INTERCEPTS ARE X=1						<BR>\n" );
document.write( "AND VERTEX IS AT X=1 AND Y=0 SO THE VERTEX IS ON THE X AXIS.						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Coordinate-system/29860: I am working with parabolas. For this problem I need to						<BR>\n" );
document.write( "Complete the square						<BR>\n" );
document.write( "Give the Vertex						<BR>\n" );
document.write( "Give the Axis						<BR>\n" );
document.write( "Give the x-intercepts						<BR>\n" );
document.write( "Give the y-intercepts						<BR>\n" );
document.write( "Give a point symmetric to the y-intercept						<BR>\n" );
document.write( "Draw the graph						<BR>\n" );
document.write( "The problem is y=x^2-2x-3.						<BR>\n" );
document.write( "Can anyone help me solve this. I am working on it and would like to have something to check my answer with.						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 16624 by venugopalramana(1167) About Me  on 2006-03-11 07:58:25 (Show Source):						<BR>\n" );
document.write( "SEE THE FOLLOWING EXAMPLE WHICH IS ALMOST SAME AS YOUR PROBLEM AND TRY.IF STILL IN DIFFICULTY PLEASE COME BACK.						<BR>\n" );
document.write( "Y=X^2-2X-3=X^2-2*X*1+1^2-1-3=(X-1)^2-1-3=(X-1)^2-4=0						<BR>\n" );
document.write( "X INTERCEPT IS OBTAINED BY PUTTING Y=0....WE GET						<BR>\n" );
document.write( "X^2-2X-3=0=X^2-3X+X-3=X(X-3)+1(X-3)=(X-3)(X+1)=0...SO..X=3 OR -1...						<BR>\n" );
document.write( "Y INTERCEPT IS GOT BY PUTTING X =0...WE GET						<BR>\n" );
document.write( "Y=0-0-3=-3						<BR>\n" );
document.write( "SEE THE GRAPH BELOW ......						<BR>\n" );
document.write( "graph( 500, 500, -10, 20, -20, 20,x^2-2x-3 )						<BR>\n" );
document.write( "THE AXIS IS X=1 AS YOU CAN SEE FROM THE GRAPH						<BR>\n" );
document.write( "ONE POINT SYMMETRIC TO Y INTERCEPT??NO...SYMMETRIC TO AXIS IT IS......(-1,3)						<BR>\n" );
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document.write( "I have to put this equation y=x^2-2x-15 into this form y=a(x-h)^2+k.						<BR>\n" );
document.write( "MAKE A PERFECT SQUARE USING X^2 AND X TERMS.ADD AND SUBTRACT THE REQUIRED CONSTANT FOR THE PURPOSE.						<BR>\n" );
document.write( "Y=(X-1)^2-1-15=(X-1)^2-16...COMPARING WITH THE ABOVE						<BR>\n" );
document.write( "y=a(x-h)^2+k.						<BR>\n" );
document.write( "WE GET A=1 AND K=-16						<BR>\n" );
document.write( "I have to find the line of symmetry.						<BR>\n" );
document.write( "X-1=0 IS THE LINE OF SYMMETRY SINCE ON EITHER SIDE OF X=1,WE GET SYMMETRIC/SAME VALUES FOR Y..AT X=1+2=3..Y IS -12 AND AT X=1-2=-1 ALSO WE GET Y=-12						<BR>\n" );
document.write( "(h,k)=vertex I think you use complete the square technique.						<BR>\n" );
document.write( "YA ..THE VERTEX AS YOU SHOULD KNOW NOW IS AT X=1 AND AT X=1 ,Y=-16 .SO (1,-16) IS THE VERTEX						<BR>\n" );
document.write( "THE GRAPH WILL LOOK LIKE THIS						<BR>\n" );
document.write( "graph( 500, 500, -10, 20, -20, 20, x^2-2x-15 )						<BR>\n" );
document.write( "Please help. thanks						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28184: What is the vertices, foci, and slope of the asymptotes for the hyperbola whose equation is, y^2/16 - x^2/25 =25?						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15970 by venugopalramana(1088) About Me  on 2006-03-04 03:12:07 (Show Source):						<BR>\n" );
document.write( "SEE THE FOLLOWING AND YOU SHOULD BE ABLE TO SOLVE YOUR PROBLEM BY YOUR SELF.....						<BR>\n" );
document.write( "THE ANSWERS FOR YOUR CASE...H=0...K=0..A=4...B=5....EQN IS OF THE TYPE						<BR>\n" );
document.write( "(Y-K)^2/B^2-(X-H)^2/A^2=1....						<BR>\n" );
document.write( "SO VERTICES ARE...(H,(K-B)) AND (H,(K+B)) ...(0,-5) AND (0,5)						<BR>\n" );
document.write( "FOCI ARE {H,(K-BE)} AND {H,(K+BE)}...WHERE E IS						<BR>\n" );
document.write( "ECCENTRICITY =SQRT{(A^2+B^2)/B^2}=SQRT((16+25)/25)=SQRT(41/25)						<BR>\n" );
document.write( "SO FOCI ARE =(0,-5SQRT(41/25) AND (0,5SQRT(41/25)...						<BR>\n" );
document.write( "OR....(0,-SQRT(41)) AND (0,SQRT(41)						<BR>\n" );
document.write( "ASYMPTOTES ARE GIVEN BY						<BR>\n" );
document.write( "Y^2/16-X^2/25=K						<BR>\n" );
document.write( "25Y^2-16X^2-400K=0						<BR>\n" );
document.write( "(5Y+4X+A)(5Y-4X+B)=0						<BR>\n" );
document.write( "SLOPES OF ASYMPTOTES ARE						<BR>\n" );
document.write( "-4/5 AND 4/5						<BR>\n" );
document.write( "THE GRAPHS LOOK LIKE THIS						<BR>\n" );
document.write( "graph( 600, 600, -10, 10, -10, 10, 4*(1+x^2/25)^0.5,-4*(1+x^2/25)^0.5,4*x/5,-4*x/5 )						<BR>\n" );
document.write( "------------------------------------------						<BR>\n" );
document.write( "What is the vertices, foci, and slope of the asymptote for the hyperbola whose equation is, y^2 - 4x^2 - 2y - 16x + 1 = 0?						<BR>\n" );
document.write( "(Y^2-2*Y*1+1^2)-{(2X)^2+2*(2X)*4+4^2}-1^2+4^2+1=0						<BR>\n" );
document.write( "(Y-1)^2-(2X+4)^2=-16						<BR>\n" );
document.write( "4(X+2)^2-(Y-1)^2=16						<BR>\n" );
document.write( "{4(X+2)^2}/16-{(Y-1)^2}/16=1						<BR>\n" );
document.write( "(X+2)^2/2^2-(Y-1)^2/4^2=1...						<BR>\n" );
document.write( "COMPARING WITH STANDARD EQN.						<BR>\n" );
document.write( "(X-H)^2/A^2-(Y-K)^2/B^2=1....WE HAVE						<BR>\n" );
document.write( "VERTICES ARE {(H-A),K} AND {(H+A),K}=(-2-2,1) AND (-2+2,1)=(-4,1) AND (0,1)						<BR>\n" );
document.write( "FOCI ARE {(H-AE),K} AND {(H+AE),K}...WHERE E IS						<BR>\n" );
document.write( "ECCENTRICITY =SQRT{(A^2+B^2)/A^2}=SQRT((4+16)/4)=SQRT(5)						<BR>\n" );
document.write( "SO FOCI ARE =(-2-2SQRT(5),1) AND (-2+2SQRT(5),1)						<BR>\n" );
document.write( "SLOPE OF ASYMPTOTE IS GIVEN BY DIFFERENTIATION.HAVE YOU BEEN TAUGHT?PLEASE INFORM.I SHALL COME BACK ON HEARING FROM YOU.						<BR>\n" );
document.write( "or you can take this proposition as proved formula						<BR>\n" );
document.write( "the pair of asymptotes for a conic is given by the same equation as the conic except for the constant term which has to be found using the condition for the equation to represent a pair of straight lines.						<BR>\n" );
document.write( "HENCE EQN OF ASYMPTOTES IS GIVEN BY						<BR>\n" );
document.write( "y^2 - 4x^2 - 2y - 16x + K=0 , WHERE K IS DETERMINED using the condition for the equation to represent a pair of straight lines.						<BR>\n" );
document.write( "SINCE WE ARE TO FIND ONLY SLOPES ,WE NEED NOT DETERMINE THE CONSTANT BUT ASSUME THAT THIS EQN REPRESENTS A PAIR OF STRAIGHT LINES.SO						<BR>\n" );
document.write( "y^2 - 4x^2 - 2y - 16x + K=0 = (Y+2X+A)(Y-2X+B)						<BR>\n" );
document.write( "HENCE SLOPES ARE +2 AND -2						<BR>\n" );
document.write( "graph( 600, 600, -10, 10, -10, 10, 1-2*((x+2)^2-4)^0.5,1+2*((x+2)^2-4)^0.5,2x+5,-2x-3 )						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28185: What is the vertices, foci, and slope of the asymptotes for the hyperbola whose equation is, x^2/81 - y^2/36 = 1?						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15969 by venugopalramana(1088) About Me  on 2006-03-04 03:09:19 (Show Source):						<BR>\n" );
document.write( "SEE THE FOLLOWING AND YOU SHOULD BE ABLE TO SOLVE YOUR PROBLEM BY YOUR SELF.....						<BR>\n" );
document.write( "THE ANSWERS FOR YOUR CASE...H=0...K=0..A=4...B=5....EQN IS OF THE TYPE						<BR>\n" );
document.write( "(Y-K)^2/B^2-(X-H)^2/A^2=1....						<BR>\n" );
document.write( "SO VERTICES ARE...(H,(K-B)) AND (H,(K+B)) ...(0,-5) AND (0,5)						<BR>\n" );
document.write( "FOCI ARE {H,(K-BE)} AND {H,(K+BE)}...WHERE E IS						<BR>\n" );
document.write( "ECCENTRICITY =SQRT{(A^2+B^2)/B^2}=SQRT((16+25)/25)=SQRT(41/25)						<BR>\n" );
document.write( "SO FOCI ARE =(0,-5SQRT(41/25) AND (0,5SQRT(41/25)...						<BR>\n" );
document.write( "OR....(0,-SQRT(41)) AND (0,SQRT(41)						<BR>\n" );
document.write( "ASYMPTOTES ARE GIVEN BY						<BR>\n" );
document.write( "Y^2/16-X^2/25=K						<BR>\n" );
document.write( "25Y^2-16X^2-400K=0						<BR>\n" );
document.write( "(5Y+4X+A)(5Y-4X+B)=0						<BR>\n" );
document.write( "SLOPES OF ASYMPTOTES ARE						<BR>\n" );
document.write( "-4/5 AND 4/5						<BR>\n" );
document.write( "THE GRAPHS LOOK LIKE THIS						<BR>\n" );
document.write( "graph( 600, 600, -10, 10, -10, 10, 4*(1+x^2/25)^0.5,-4*(1+x^2/25)^0.5,4*x/5,-4*x/5 )						<BR>\n" );
document.write( "------------------------------------------						<BR>\n" );
document.write( "What is the vertices, foci, and slope of the asymptote for the hyperbola whose equation is, y^2 - 4x^2 - 2y - 16x + 1 = 0?						<BR>\n" );
document.write( "(Y^2-2*Y*1+1^2)-{(2X)^2+2*(2X)*4+4^2}-1^2+4^2+1=0						<BR>\n" );
document.write( "(Y-1)^2-(2X+4)^2=-16						<BR>\n" );
document.write( "4(X+2)^2-(Y-1)^2=16						<BR>\n" );
document.write( "{4(X+2)^2}/16-{(Y-1)^2}/16=1						<BR>\n" );
document.write( "(X+2)^2/2^2-(Y-1)^2/4^2=1...						<BR>\n" );
document.write( "COMPARING WITH STANDARD EQN.						<BR>\n" );
document.write( "(X-H)^2/A^2-(Y-K)^2/B^2=1....WE HAVE						<BR>\n" );
document.write( "VERTICES ARE {(H-A),K} AND {(H+A),K}=(-2-2,1) AND (-2+2,1)=(-4,1) AND (0,1)						<BR>\n" );
document.write( "FOCI ARE {(H-AE),K} AND {(H+AE),K}...WHERE E IS						<BR>\n" );
document.write( "ECCENTRICITY =SQRT{(A^2+B^2)/A^2}=SQRT((4+16)/4)=SQRT(5)						<BR>\n" );
document.write( "SO FOCI ARE =(-2-2SQRT(5),1) AND (-2+2SQRT(5),1)						<BR>\n" );
document.write( "SLOPE OF ASYMPTOTE IS GIVEN BY DIFFERENTIATION.HAVE YOU BEEN TAUGHT?PLEASE INFORM.I SHALL COME BACK ON HEARING FROM YOU.						<BR>\n" );
document.write( "or you can take this proposition as proved formula						<BR>\n" );
document.write( "the pair of asymptotes for a conic is given by the same equation as the conic except for the constant term which has to be found using the condition for the equation to represent a pair of straight lines.						<BR>\n" );
document.write( "HENCE EQN OF ASYMPTOTES IS GIVEN BY						<BR>\n" );
document.write( "y^2 - 4x^2 - 2y - 16x + K=0 , WHERE K IS DETERMINED using the condition for the equation to represent a pair of straight lines.						<BR>\n" );
document.write( "SINCE WE ARE TO FIND ONLY SLOPES ,WE NEED NOT DETERMINE THE CONSTANT BUT ASSUME THAT THIS EQN REPRESENTS A PAIR OF STRAIGHT LINES.SO						<BR>\n" );
document.write( "y^2 - 4x^2 - 2y - 16x + K=0 = (Y+2X+A)(Y-2X+B)						<BR>\n" );
document.write( "HENCE SLOPES ARE +2 AND -2						<BR>\n" );
document.write( "graph( 600, 600, -10, 10, -10, 10, 1-2*((x+2)^2-4)^0.5,1+2*((x+2)^2-4)^0.5,2x+5,-2x-3 )						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28183: What is the center, foci, and the lengths of the major and minor axes for the						<BR>\n" );
document.write( "ellipse, 16x^2 + 25y^2 + 32x - 150y = 159?						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15968 by venugopalramana(1088) About Me  on 2006-03-04 03:04:16 (Show Source):						<BR>\n" );
document.write( "SEE THE FOLLOWING EXAMPLE						<BR>\n" );
document.write( "What is the center, foci, and the length of the major and minor axes for the ellipse whose equation is, 16x^2 + 25y^2 + 32x - 150y = 159?						<BR>\n" );
document.write( "STANDARD EQN.OF ELLIPSE IS						<BR>\n" );
document.write( "(X-H)^2/A^2 + (Y-K)^2/B^2 = 1						<BR>\n" );
document.write( "WHERE						<BR>\n" );
document.write( "CENTRE IS (H,K)						<BR>\n" );
document.write( "ECCENTRICITY = E = {(A^2-B^2)/A^2}^0.5						<BR>\n" );
document.write( "FOCI ARE (H+AE,K)AND (H-AE,K)						<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2A						<BR>\n" );
document.write( "MINOR AXIS LENGTH = 2B						<BR>\n" );
document.write( "WE HAVE						<BR>\n" );
document.write( "16x^2 + 25y^2 + 32x - 150y - 159 =0						<BR>\n" );
document.write( "{(4X)^2+2*(4X)*4+4^2}-4^2+{(5Y)^2-2*(5Y)*15+15^2}-15^2-159=0						<BR>\n" );
document.write( "(4X+4)^2 + (5Y-15)^2 = 400						<BR>\n" );
document.write( "16(X+1)^2 + 25(Y-3)^2 =400...DIVIDING BY 400 THROUGHOUT..						<BR>\n" );
document.write( "(X+1)^2/25 + (Y-3)^2/16 =1						<BR>\n" );
document.write( "(X+1)^2/5^2 + (Y-3)^2/4^2 =1						<BR>\n" );
document.write( "COMPARING WITH ABOVE STANDARD EQN.						<BR>\n" );
document.write( "CENTRE IS (-1,3)						<BR>\n" );
document.write( "ECCENTRICITY E IS {(25-16)/25}^0.5=3/5						<BR>\n" );
document.write( "FOCI ARE (-1+5*3/5 ,3) AND (-1-5*3/5,3)=(2,3) AND (-4,-3)						<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2*5=10						<BR>\n" );
document.write( "MINOR AXIS LENGTH = 2*4=8						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28182: What is the center, foci, and the lengths of the major and minor axes for the						<BR>\n" );
document.write( "ellipse, (x-4)^2/16 + (y+1)^2/9 = 1?						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15966 by venugopalramana(1088) About Me  on 2006-03-04 03:01:34 (Show Source):						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "SEE THE FOLLOWING EXAMPLE						<BR>\n" );
document.write( "What is the center, foci, and the length of the major and minor axes for the ellipse whose equation is, 16x^2 + 25y^2 + 32x - 150y = 159?						<BR>\n" );
document.write( "STANDARD EQN.OF ELLIPSE IS						<BR>\n" );
document.write( "(X-H)^2/A^2 + (Y-K)^2/B^2 = 1						<BR>\n" );
document.write( "WHERE						<BR>\n" );
document.write( "CENTRE IS (H,K)						<BR>\n" );
document.write( "ECCENTRICITY = E = {(A^2-B^2)/A^2}^0.5						<BR>\n" );
document.write( "FOCI ARE (H+AE,K)AND (H-AE,K)						<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2A						<BR>\n" );
document.write( "MINOR AXIS LENGTH = 2B						<BR>\n" );
document.write( "WE HAVE						<BR>\n" );
document.write( "16x^2 + 25y^2 + 32x - 150y - 159 =0						<BR>\n" );
document.write( "{(4X)^2+2*(4X)*4+4^2}-4^2+{(5Y)^2-2*(5Y)*15+15^2}-15^2-159=0						<BR>\n" );
document.write( "(4X+4)^2 + (5Y-15)^2 = 400						<BR>\n" );
document.write( "16(X+1)^2 + 25(Y-3)^2 =400...DIVIDING BY 400 THROUGHOUT..						<BR>\n" );
document.write( "(X+1)^2/25 + (Y-3)^2/16 =1						<BR>\n" );
document.write( "(X+1)^2/5^2 + (Y-3)^2/4^2 =1						<BR>\n" );
document.write( "COMPARING WITH ABOVE STANDARD EQN.						<BR>\n" );
document.write( "CENTRE IS (-1,3)						<BR>\n" );
document.write( "ECCENTRICITY E IS {(25-16)/25}^0.5=3/5						<BR>\n" );
document.write( "FOCI ARE (-1+5*3/5 ,3) AND (-1-5*3/5,3)=(2,3) AND (-4,-3)						<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2*5=10						<BR>\n" );
document.write( "MINOR AXIS LENGTH = 2*4=8						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28745: Hi, In my Algebra II class we are working on Solving Quadratic Equations by graphing. I relize how to work the problems with a plain vertex like in this problem x2+2x-8=0 the vertex is 1 and so the root/answer would be (-4 and 2) but i am unsure of solving the ones with a fraction for the vertex like in this problem x2-5x+4=0 and the vertex is 5/2 could you please help me?						<BR>\n" );
document.write( "thanks so much,						<BR>\n" );
document.write( "Chelsea						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15712 by venugopalramana(1088) About Me  on 2006-02-28 01:31:49 (Show Source):						<BR>\n" );
document.write( "Hi, In my Algebra II class we are working on Solving Quadratic Equations by graphing. I relize how to work the problems with a plain vertex like in this problem x2+2x-8=0						<BR>\n" );
document.write( "(X+1)^2-9=0 = (X+1)^2- (3)^2						<BR>\n" );
document.write( "the vertex is 1 ....NO....PLEASE CORRECT						<BR>\n" );
document.write( "THE VERTEX IS AT X=-1 AND Y = -9....						<BR>\n" );
document.write( "and so the root/answer would be (-4 and 2)...THAT IS -1+3=2 AND -1-3=-4						<BR>\n" );
document.write( "OK...GOOD..SEE THE GRAPH BELOW...						<BR>\n" );
document.write( "but i am unsure of solving the ones with a fraction for the vertex like in this problem x2-5x+4=0						<BR>\n" );
document.write( "(X-5/2)^2 - 9/4 = 0 = (X-5/2)^2 - (3/2)^2						<BR>\n" );
document.write( "and the vertex is 5/2						<BR>\n" );
document.write( "CORRECT ..YOU ARE CORRECT HERE..HOW DID YOU MISTAKE IN THE EARLIER CASE?						<BR>\n" );
document.write( "could you please help me?						<BR>\n" );
document.write( "SO THE ROOTS ARE 5/2 + 3/2 =8/2 = 4.......OR.....5/2 - 3/2 = 2/2 =1...THAT IS 4 OR 1.						<BR>\n" );
document.write( "I HOPE YOU GOT THE METHOD..THE ROOTS ARE OBTAINED BY ADDING /SUBTRACTING THE SQUARE ROOT OF CONSTANT TERM FROM THE VERTEX (OR PRECISELY X COORDINATE OF VERTEX)						<BR>\n" );
document.write( "SEE THE GRAPH BELOW						<BR>\n" );
document.write( "graph( 600, 600, -10, 10, -10, 10,-1+(x+9)^0.5,-1-(x+9)^0.5,2.5+(x+2.25)^0.5,2.5-(x+2.25)^0.5,-1,2.5 )						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28744: I have a parabola that I need to find an equation for.						<BR>\n" );
document.write( "so far all i have is y=negative___x squared+6. I need to fill in the blank.						<BR>\n" );
document.write( "I know it passes through (neg.4,5) and (7,2)						<BR>\n" );
document.write( "help me figure out what goes in the blank please						<BR>\n" );
document.write( "I just cannot remember how to do it out						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15711 by venugopalramana(1088) About Me  on 2006-02-28 01:18:17 (Show Source):						<BR>\n" );
document.write( "I have a parabola that I need to find an equation for.						<BR>\n" );
document.write( "so far all i have is y=negative___x squared+6. I need to fill in the blank.						<BR>\n" );
document.write( "I know it passes through (neg.4,5) and (7,2)						<BR>\n" );
document.write( "help me figure out what goes in the blank please						<BR>\n" );
document.write( "I just cannot remember how to do it out						<BR>\n" );
document.write( "Y=-A*X^2+6...IF IT PASSES THROUGH (-4,5)						<BR>\n" );
document.write( "5= -A *(-4)^2+6 = -16A+6						<BR>\n" );
document.write( "16A=6-5=1						<BR>\n" );
document.write( "A=1/16.....						<BR>\n" );
document.write( "LET US CHECK WHETHER (7,2) LIES ON THAT						<BR>\n" );
document.write( "2=-(1/16)*7^2+6 IS NOT CORRECT ..SO PLEASE CHECK YOUR DATA						<BR>\n" );
document.write( "BUT THIS IS THE METHOD OF SOLUTION.SINCE THERE IS ONLY 1 UNKNOWN WE NEED ONLY ONE POINT..THE SECOND POINT IS REDUNDANT..						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28461: What is the center, foci, and the length of the major and minor axes for the ellipse whose equation is, (x-4)^2/16 + (y+1)^2/9 = 1?						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15710 by venugopalramana(1088) About Me  on 2006-02-28 01:11:31 (Show Source):						<BR>\n" );
document.write( "SEE THE FOLLOWING EXAMPLE AND TRY. IF STILL IN DIFFICULTY PLEASE COME BACK.						<BR>\n" );
document.write( "------------------------------------------------------------------------------						<BR>\n" );
document.write( "What is the center, foci, and the length of the major and minor axes for the ellipse whose equation is, 16x^2 + 25y^2 + 32x - 150y = 159?						<BR>\n" );
document.write( "STANDARD EQN.OF ELLIPSE IS						<BR>\n" );
document.write( "(X-H)^2/A^2 + (Y-K)^2/B^2 = 1						<BR>\n" );
document.write( "WHERE						<BR>\n" );
document.write( "CENTRE IS (H,K)						<BR>\n" );
document.write( "ECCENTRICITY = E = {(A^2-B^2)/A^2}^0.5						<BR>\n" );
document.write( "FOCI ARE (H+AE,K)AND (H-AE,K)						<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2A						<BR>\n" );
document.write( "MINOR AXIS LENGTH = 2B						<BR>\n" );
document.write( "WE HAVE						<BR>\n" );
document.write( "16x^2 + 25y^2 + 32x - 150y - 159 =0						<BR>\n" );
document.write( "{(4X)^2+2*(4X)*4+4^2}-4^2+{(5Y)^2-2*(5Y)*15+15^2}-15^2-159=0						<BR>\n" );
document.write( "(4X+4)^2 + (5Y-15)^2 = 400						<BR>\n" );
document.write( "16(X+1)^2 + 25(Y-3)^2 =400...DIVIDING BY 400 THROUGHOUT..						<BR>\n" );
document.write( "(X+1)^2/25 + (Y-3)^2/16 =1						<BR>\n" );
document.write( "(X+1)^2/5^2 + (Y-3)^2/4^2 =1						<BR>\n" );
document.write( "COMPARING WITH ABOVE STANDARD EQN.						<BR>\n" );
document.write( "CENTRE IS (-1,3)						<BR>\n" );
document.write( "ECCENTRICITY E IS {(25-16)/25}^0.5=3/5						<BR>\n" );
document.write( "FOCI ARE (-1+5*3/5 ,3) AND (-1-5*3/5,3)=(2,3) AND (-4,-3)						<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2*5=10						<BR>\n" );
document.write( "MINOR AXIS LENGTH = 2*4=8						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "						<BR>\n" );
document.write( "Quadratic-relations-and-conic-sections/28463: What is the center, foci, and the length of the major and minor axes for the ellipse whose equation is, 16x^2 + 25y^2 + 32x - 150y = 159?						<BR>\n" );
document.write( "1 solutions						<BR>\n" );
document.write( "Answer 15709 by venugopalramana(1088) About Me  on 2006-02-28 01:03:38 (Show Source):						<BR>\n" );
document.write( "What is the center, foci, and the length of the major and minor axes for the ellipse whose equation is, 16x^2 + 25y^2 + 32x - 150y = 159?						<BR>\n" );
document.write( "STANDARD EQN.OF ELLIPSE IS						<BR>\n" );
document.write( "(X-H)^2/A^2 + (Y-K)^2/B^2 = 1						<BR>\n" );
document.write( "WHERE						<BR>\n" );
document.write( "CENTRE IS (H,K)						<BR>\n" );
document.write( "ECCENTRICITY = E = {(A^2-B^2)/A^2}^0.5						<BR>\n" );
document.write( "FOCI ARE (H+AE,K)AND (H-AE,K)						<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2A						<BR>\n" );
document.write( "MINOR AXIS LENGTH = 2B						<BR>\n" );
document.write( "WE HAVE						<BR>\n" );
document.write( "16x^2 + 25y^2 + 32x - 150y - 159 =0						<BR>\n" );
document.write( "{(4X)^2+2*(4X)*4+4^2}-4^2+{(5Y)^2-2*(5Y)*15+15^2}-15^2-159=0						<BR>\n" );
document.write( "(4X+4)^2 + (5Y-15)^2 = 400						<BR>\n" );
document.write( "16(X+1)^2 + 25(Y-3)^2 =400...DIVIDING BY 400 THROUGHOUT..						<BR>\n" );
document.write( "(X+1)^2/25 + (Y-3)^2/16 =1						<BR>\n" );
document.write( "(X+1)^2/5^2 + (Y-3)^2/4^2 =1						<BR>\n" );
document.write( "COMPARING WITH ABOVE STANDARD EQN.						<BR>\n" );
document.write( "CENTRE IS (-1,3)						<BR>\n" );
document.write( "ECCENTRICITY E IS {(25-16)/25}^0.5=3/5						<BR>\n" );
document.write( "FOCI ARE (-1+5*3/5 ,3) AND (-1-5*3/5,3)=(2,3) AND (-4,-3)								<BR>\n" );
document.write( "MAJOR AXIS LENGTH = 2*5=10								<BR>\n" );
document.write( "Linear_Algebra/30362: Question: Find the equation of the ellipse whose center is (5,-3) that has a vertex at 13,-3) and a minor axis of lenght 10. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );