SOLUTION: Please help me with this! Find the centre, vertices, lenght of latus rectum, foci , eccentricities of the ellipse x^2 + 9y^2 + 4x -18y - 23 = 0

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Please help me with this! Find the centre, vertices, lenght of latus rectum, foci , eccentricities of the ellipse x^2 + 9y^2 + 4x -18y - 23 = 0      Log On


   

Question 991825: Please help me with this! Find the centre, vertices, lenght of latus rectum, foci , eccentricities of the ellipse x^2 + 9y^2 + 4x -18y - 23 = 0
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
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x^2 + 9y^2 + 4x -18y - 23 = 0
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Complete the squares:

x%5E2+%2B+9y%5E2+%2B+4x+-18y+-+23 = %28x%2B2%29%5E2+-+4+%2B+%283y-3%29%5E2+-+9+-+23 = %28x%2B2%29%5E2+%2B+3%5E2%2A%28y-1%29%5E2+-+36.

So,  you have the equation

%28x%2B2%29%5E2 + 9%2A%28y-1%29%5E2%29 = 36,

or,  in the canonical form,

%28x%2B2%29%5E2%2F6%5E2 + %28%28y-1%29%5E2%29%2F%282%5E2%29 = 1.

It is a canonical equation of an ellipse.

Its center is at the point  (x,y) = (-2,1).  Semi-major and semi-minor axes are  6  and  2,  respectively:  a=6  and  b=2.
Semi-major axis is the distance from the ellipse' center to its vertices.

Linear eccentricity is  sqrt%286%5E2+-+2%5E2%29 = sqrt%2832%29 = 4%2Asqrt%282%29.  It is the distance from the ellipse's center to its foci.

Next,  take the textbook and determine the rest ellipse's characteristics you need.