Find the Center (x^2)/400+(y^2)/256=1
The question is asking you to determine the center of an ellipse represented by the equation \((x^2)/400 + (y^2)/256 = 1\). In general, the standard form of the equation of an ellipse centered at \((h, k)\) is \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\). The question requires you to identify the coordinates \(h\) and \(k\) which describe the location of the center of this particular ellipse within the Cartesian coordinate system.
$\frac{x^{2}}{400} + \frac{y^{2}}{256} = 1$
Answer
Solution:
Step 1:
Rewrite the given equation to match the standard form of an ellipse or hyperbola, which requires the equation to have $1$ on the right-hand side. The equation is already in the correct form: $\frac{x^{2}}{400} + \frac{y^{2}}{256} = 1$.
Step 2:
Recognize that the equation represents an ellipse. The standard equation for an ellipse is $\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1$, where $(h,k)$ is the center, $a$ is the semi-major axis length, and $b$ is the semi-minor axis length.
Step 3:
Compare the given equation to the standard form to find the values corresponding to $a$, $b$, $h$, and $k$. In this case, $a = 20$, $b = 16$, and since there is no $h$ or $k$ in the equation, $h = 0$ and $k = 0$.
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:
The center of the ellipse is the point $(0, 0)$.
Knowledge Notes:
The standard form of an ellipse is $\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1$, where:
$(h, k)$ is the center of the ellipse.
$a$ is the length of the semi-major axis, which is the longest radius of the ellipse.
$b$ is the length of the semi-minor axis, which is the shortest radius of the ellipse.
If $a > b$, the major axis is horizontal; if $b > a$, the major axis is vertical.
In the given problem, the equation is already in the standard form, with $a$ and $b$ being the square roots of the denominators of the $x^2$ and $y^2$ terms, respectively. Since there are no $(x - h)$ or $(y - k)$ terms, it implies that $h = 0$ and $k = 0$, indicating that the center of the ellipse is at the origin of the coordinate system.
The center of an ellipse is an important characteristic because it defines the position of the ellipse in the coordinate plane. The lengths of the semi-major and semi-minor axes determine the shape and size of the ellipse.
