91Ó°ÊÓ

To find the center of an ellipse, we must identify specific values from its equation in standard form. The standard equation is \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \). Here, the center is denoted by \((h, k)\). It is the midpoint of the ellipse where both the major and minor axes intersect.

Let's compare it to the provided ellipse equation: \( \frac{(x+2)^{2}}{16} + (y-3)^{2}=1 \). Upon comparison, we can see that \( x+2 \) and \( y-3 \) represent our \( (x-h) \) and \( (y-k) \) parts, respectively.

This means that \( h = -2 \) and \( k = 3 \). Therefore, the center of the ellipse in this case is \((-2, 3)\). This center acts as the anchor point for determining the overall structure of the ellipse.

The major and minor axes are crucial components in understanding the geometry of an ellipse. They determine the shape and size of the ellipse.

• The **major axis** is the longest diameter of the ellipse. It spans the full width and passes through both foci. In the standard equation \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \), the term under the larger denominator \((a^{2} > b^{2})\) represents the length of the major axis.


• The **minor axis** is the shortest diameter. It is perpendicular to the major axis at the center of the ellipse. In our equation, the smaller denominator term \(b^{2}\) corresponds to the minor axis length.

In the equation \( \frac{(x+2)^{2}}{16} + (y-3)^{2} = 1 \), we've identified \( a^{2} = 16 \) and \( b^{2} = 1 \).

Calculating, we find:

• \( a = \sqrt{16} = 4 \)
• \( b = \sqrt{1} = 1 \)

So, the **lengths** of the axes double these values:

• Major axis length = \(2a = 8\)
• Minor axis length = \(2b = 2\)

The major axis runs parallel to the x-axis, and the minor axis is along the y-axis.

The ellipse is a geometric shape defined by a specific type of equation. The standard form of the ellipse equation is essential for identifying and sketching the ellipse.

This form is \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \).

Breaking it down:

• \((h, k)\) represents the **center** of the ellipse.
• \(a^{2}\) and \(b^{2}\) are the **denominators** indicating the square of the distances from the center to the ellipse along the x-axis and y-axis, respectively.
• If \(a^{2} > b^{2}\), the major axis is horizontal, and if \(b^{2} > a^{2}\), it's vertical.

The given problem uses this standard form and inverts the signs inside the parentheses, indicating a translation from the origin to the center \((-h, -k)\).

By working with this format, you can easily **graph** an ellipse and understand its orientation and dimensions. The equation gives us a 'recipe' for plotting the ellipse precisely on a coordinate plane.