To find the equation of an ellipse given the distance between the foci, the minor axis length, and the focal width, we can use the standard form of the equation for an ellipse centered at the origin:
(x²/a²) + (y²/b²) = 1
where 'a' represents the semi-major axis and 'b' represents the semi-minor axis.
In this case, the distance between the foci equals the minor axis length. Let's denote the minor axis length as '2b', so 'b' represents half of the minor axis length.
Given that the focal width is 4, we know that the distance between the foci is 2c = 4, where 'c' represents the distance between the center and each focus. Since 'c' is half the focal width, we have c = 2.
To find 'a', we can use the relationship between 'a', 'b', and 'c' in an ellipse:
a² = b² + c²
Substituting the known values, we have:
a² = (2b)² + (2)² a² = 4b² + 4 a² = 4(b² + 1)
Now, let's substitute 'a' and 'b' into the standard form equation:
(x²/(4(b² + 1))) + (y²/b²) = 1
Simplifying the equation, we get:
(x²/(4b² + 4)) + (y²/b²) = 1
Therefore, the equation of the ellipse, with the given conditions, is:
x²/(4b² + 4) + y²/b² = 1