How do you write 25y^2 + 9x^2 - 50y - 54x = 11925y2+9x2−50y−54x=119 in standard form?

1 Answer
Alan P.
May 3, 2015

25y^2+9x^2-50y-54x = 11925y2+9x2−50y−54x=119

Consider the tow sub-expressions from the left side of this equation:

  1. Terms involving color(red)(y)y
    color(red)(25y^2-50y)25y2−50y
    color(red)(= 25(y^2-2y))=25(y2−2y)
    color(red)(=25(y^2-2y+1) -25)=25(y2−2y+1)−25
    color(red)(=5^2(y-1)^2 -25)=52(y−1)2−25

  2. Terms involving color(blue)(x)x
    color(blue)(9x^2-54x)9x2−54x
    color(blue)(=9(x^2-6x+(-3)^2) -81)=9(x2−6x+(−3)2)−81
    color(blue)(=3^2(x-3)^2 -81)=32(x−3)2−81

25y^2+9x^2-50y-54x = 11925y2+9x2−50y−54x=119
25y^2-50y +9x^2-54x = 11925y2−50y+9x2−54x=119
=color(red)(5^2(y-1)^2 -25) + color(blue)(3^2(x-3)^2-81) = 119=52(y−1)2−25+32(x−3)2−81=119
=5^2(y-1)^2+3^2(x-3)^2 = 225=52(y−1)2+32(x−3)2=225
or
=(25y-25)^2+(9x-27)^2= 15^2=(25y−25)2+(9x−27)2=152
or
some variant of this depending upon your local definition of "standard form" for an ellipse.

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