Given the ellipse 25x^2 + 5y^2 + 50x - 20y + 20 = 0 find: a) The Center C b) The Length of the Major Axis c) The Length of the Minor Axis d) Distance from C to foci
The standard forms of horizontal and vertical ellipses are respectively:
`(x-h)^2/a^2+(y-k)^2/b^2=1` and` (x-h)^2/b^2+(y-k)^2/a^2=1` where (h,k) is the center, a is semi major axis, b is semi minor axis.
The given equation is `25x^2 + 5y^2 + 50x - 20y + 20 = 0`
Rewrite the equation in the form:
`25x^2 + 50x+25+5y^2 -20y+20=-20+25+20 `
Dividing both sides by 25
This is the standard equation of a vertical ellipse.
1) Therefore,the Center C is (-1,2)
2) To determine the length of the major axis, consider the larger denominator which is 5.
length of major axis =`2a=2*sqrt5=2sqrt5` units.
3) To determine the length of the minor axis, consider the smaller denominator which is 1.
length of the minor axis = 2b=2*1=2 units.
4) To find the distance, c of the foci from the center of the ellipse, apply the formula:
Plugging in the values of `a^2` and `b^2`
`C=sqrt(5-1)` =2 units
Hence, the distance of each focus from the center of the ellipse is 2 units.