Question

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

Answer 1

`25y^2+9x^2-50y-54x = 119`

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

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

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

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