Find the Center (x^2)/81+(y^2)/225=1
This problem is asking for the center of an ellipse represented by the equation (x^2)/81 + (y^2)/225 = 1. An ellipse is a geometric shape that looks like an elongated circle. The standard form of an ellipse's equation is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively. The question requires identifying the coordinates (h, k) that correspond to the center of the ellipse based on the given equation.
$\frac{x^{2}}{81} + \frac{y^{2}}{225} = 1$
Answer
Solution:
Step 1:
Rewrite the given equation to match the standard form of an ellipse equation, ensuring the right side equals $1$. The equation is already in the required form: $\frac{x^{2}}{81} + \frac{y^{2}}{225} = 1$.
Step 2:
Identify the equation as representing an ellipse. The standard equation for an ellipse is $\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively, and $(h, k)$ is the center of the ellipse.
Step 3:
Compare the given equation with the standard form to find the values corresponding to $a$, $b$, $h$, and $k$. In this case, we find $a = 15$, $b = 9$, and since there is no $h$ or $k$ present, they are both $0$.
Step 4:
Determine the center of the ellipse using the coordinates $(h, k)$. Plugging in the values for $h$ and $k$, we get the center at $(0, 0)$.
Step 5:
Conclude that the center of the ellipse described by the equation $\frac{x^{2}}{81} + \frac{y^{2}}{225} = 1$ is at the origin, which is the point $(0, 0)$.
Knowledge Notes:
The problem involves finding the center of an ellipse. The relevant knowledge points include:
Standard Form of an Ellipse: The standard form of an ellipse with horizontal and vertical axes is given by $\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1$, where $(h, k)$ is the center of the ellipse, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis.
Center of an Ellipse: The center of an ellipse in the standard form is the point $(h, k)$. If the equation does not have $x$ or $y$ shifted by any value, then $h$ and $k$ are both $0$, indicating the center is at the origin.
Major and Minor Axes: In the standard form, $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively. If $a > b$, the ellipse is elongated along the x-axis, and if $b > a$, it is elongated along the y-axis.
Simplification: Before identifying the center, it is important to ensure the equation of the ellipse is simplified so that the right side of the equation equals $1$. This allows for easy comparison with the standard form.
Understanding these concepts is crucial for solving problems related to the geometry of ellipses, including finding the center, foci, lengths of axes, and graphing the ellipse.
