Identify the type of conic section that has the equation 9x^2 + 25y^2 =225 and identify its domain and range.
The + on the left means it's a circle or ellipse.
The differing coefficients for x^2 and y^2 means it's an ellipse.
In standard form, that is just
x^2/25 + y^2/9 = 1
Whats the domain and range?
Domain:
[
−
4
,
4
]
,
{
x
|
−
4
≤
x
≤
4
}
[
-
4
,
4
]
,
{
x
|
-
4
≤
x
≤
4
}
Range:
[
−
3
,
3
]
,
{
y
|
−
3
≤
y
≤
3
}
To identify the type of conic section represented by the equation 9x^2 + 25y^2 = 225, we need to examine the coefficients of x^2 and y^2.
In this equation, the coefficient of x^2 is positive (9) and the coefficient of y^2 is also positive (25). Since both coefficients are positive and equal, we can conclude that the conic section is an ellipse.
To determine the domain and range of the ellipse, we need to rewrite the equation in a form that allows us to identify the major and minor axes. Here's how we can do that:
Divide the equation by 225 to simplify it:
(x^2) / (25/9) + (y^2) / (225/25) = 1
Now, we can rewrite it as follows:
(x^2) / (5/3)^2 + (y^2) / (15/5)^2 = 1
Comparing this equation to the standard equation of an ellipse in the form:
(x^2) / (a^2) + (y^2) / (b^2) = 1
We can see that a = 5/3 and b = 15/5.
Since a represents the radius of the ellipse in the x-direction and b represents the radius in the y-direction, we can determine the domain and range.
The domain refers to the possible values of x, while the range refers to the possible values of y.
For this ellipse, since the coefficient of x^2 is larger than the coefficient of y^2, the major axis is aligned with the x-axis. Consequently, we can determine the range from the value of b.
The range (values of y) for this ellipse is given by:
Range: [-15/5, 15/5] or [-3, 3]
Since the minor axis is aligned with the y-axis for this ellipse, we can determine the domain from the value of a.
The domain (values of x) for this ellipse is given by:
Domain: [-5/3, 5/3] or [-1.67, 1.67]
So, the domain of this ellipse is [-5/3, 5/3] and the range is [-3, 3].