Given the equation 4x^2 + y^2 - 48x -4y + 48 = 0 find:

a) The Center C

b) Length of Major Axis

c) Length of Minor Axis

d) Distance from C to foci

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Given equation is:

`4x^2 + y^2 - 48x -4y + 48 = 0`

Rewrite the equation in the form:

`4x^2- 48x + 144 + y^2 -4y+4=-48 +144+4`

`rArr 4(x^2-12x+36) + y^2 -4y+4=100`

`rArr 4(x-6)^2 + (y-2)^2=100`

Divide both sides by 100,

`(x-6)^2/5^2 + (y-2)^2/10^2=1`

This is the standard equation of a vertical ellipse.

1. Its center C is at (6,2)

2. To determine the length of the major axis, consider the larger denominator which is 100.

So, `a^2=10^2`

`rArr a=10`

length of major axis = 2a=2*10*=***20 units**.

3. Similarly, `b^2=5^2`

`rArr b=5`

length of minor axis = 2b=2*5=**10 units.**

4. To find the distance of the foci from the center of the ellipse, apply the formula:

`c=sqrt(a^2-b^2)`

Plugging in the values of `a^2` and `b^2`

c=`sqrt(100-25)` =`sqrt(75)` =`5sqrt3` units

**Hence, the distance of each focus from the center of the ellipse is ****`5sqrt3` units .**

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