Can you get an equation of an ellipse from knowing only the foci?

Unfortunately, the answer is in the negative. Imagine two pins placed at the given foci, and a string, attached at each end, going between the pins. For any length of string greater than the distance between the pins, there is an ellipse with the given foci. All of the ellipses will have the same center (12, 0), but not necessarily the same eccentricity. Follow the link below to see a graph on photobucket.

the distance between the foci is 2c = 15 -9 = 6

therefore c = 3

and you have a^2 = b^2 + c^2 = b^2 + 9

b^2 = a^2 - 9

the center of ellipse is ( 12, 0 )

the equation therefore is

(x- 12 )^ 2/ (a^2 ) + y^2 / ( a^2 - 9 ) = 1

where a is a parameter.

There will be an indefinite number of ellipses.

The restriction of a is a^2 - 9 > 0

or a >3

For example when a = 4

then the equation of ellipse is

(x-12)^2 / 16 + y^2 / 7 = 1

when a= 3 the term y^2 / ( a^2 -9 ) goes to infinity

and when a < 3 you have a hyperbola which is irrelevant here.

What if the question

Given that an ellipse passing through (3,2) shares the same foci of an ellipse x^2/9 + y^2/4:1.Find the equation of an ellipse? Can you help me with this??

No, that is not enough information. The foci give you the orientation of the ellipse (horizontal), the center point (12, 0), and the value of c (3), but don't tell you how far it extends horizontally and vertically. You need the values of a and b for that.

c² = a² + b²

You have only one equation for two unknowns.

factor on ellipse is (-8,-3) ordinary equation of an ellipse is x²/a² +y²/b² =a million (-8)²/a²+(-3)²/b²=a million sixty 4/a²+9/b²=a million 64b² +9a² =a²b² 64b² -a²b²=-9a² b² = -9a² / [ sixty 4-a² sixty 4-a² ] ------------- c²=a²-b² 3² = a² -b² b²=a²-9 a²-9 = -9a² / [ sixty 4-a² sixty 4-a² ] (a²-9)(sixty 4-a²) =-9a² -a^4 +73a² -576 =-9a² a^4 -82a²+576=0 a²=t t²-82t+576=0 t= 80 two±?4420 / 2 => 40-one±?1105 a²= 40-one+?1105 b²=a²-9 => 32+?1105 equation of the ellipse is : x² / 40-one+?1105 + y² / 32+?1105 =a million tell me what are the possibilities? perhaps we are able to alter our equation by skill of substracting or multiplying.

No, you can't.