Question

Find the coordinates of the center, vertices, and foci for each ellipse. Round to three significant digits where needed.$$\frac{(x+5)^{2}}{25}+\frac{(y-3)^{2}}{49}=1$$

Short answer

The center of the ellipse is (-5, 3), the vertices are (-5, -4) and (-5, 10), and the foci are approximately (-5, -1.899) and (-5, 7.899), rounded to three significant digits.

Step-by-step solution

Identify the Standard Form of the Ellipse Equation

The given ellipse equation is already in standard form, where the standard form is \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\) for an ellipse centered at (h, k) with semi-major axis length 'a' aligned along the x-axis and semi-minor axis length 'b' aligned along the y-axis, or to flip the axes if \(a^{2} < b^{2}\).

Determine the Center of the Ellipse

The center (h, k) of the ellipse can be determined by the values inside the parentheses of the equation. Since the equation is \(\frac{(x+5)^{2}}{25} + \frac{(y-3)^{2}}{49} = 1\), the center is at (-5, 3).

Find the Lengths of the Semi-Major and Semi-Minor Axes

In this equation, \(\frac{(x+5)^{2}}{25}\) and \(\frac{(y-3)^{2}}{49}\), the denominators represent the squares of the semi-major and semi-minor axes. Since \(25 < 49\), \(a^{2} = 25\) (with \(a = 5\)) is the length of the semi-minor axis and \(b^{2} = 49\) (with \(b = 7\)) is the length of the semi-major axis.

Identify the Vertices

Vertices are located \(a\) units from the center along the major axis. With the center being (-5, 3), the vertices are at (-5, 3±b) or (-5, 3±7). Hence, the vertices are (-5, -4) and (-5, 10).

Calculate the Distance of the Foci from the Center

The distance of the foci from the center is given by \(c\), where \(c^{2} = b^{2} - a^{2}\). Here, \(c^{2} = 49 - 25 = 24\), thus \(c = \sqrt{24} \approx 4.899\).

Locate the Foci

The foci are c units away from the center along the major axis. Therefore, they are located at (-5, 3±c) or (-5, 3±4.899). The approximate coordinates of the foci, rounded to three significant digits, are (-5, -1.899) and (-5, 7.899).

Key concepts

Ellipse Standard Form

Understanding the standard form of an ellipse is crucial for analyzing and graphing it. The standard form equation for an ellipse is \[\begin{equation}\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\end{equation}\] if the ellipse is horizontally oriented, with the larger axis lying along the x-axis. If the ellipse is vertically oriented, the equation is the same, but with the roles of 'a' and 'b' switched. Here,

• (h, k) represent the coordinates of the center of the ellipse.
• \(a\) is the length of the semi-major axis, and
• \(b\) is the length of the semi-minor axis.

The standard form makes it clear where the ellipse is centered and the lengths of its axes. This organization aids in predicting the shape and position of the ellipse before plotting it graphically. For the given equation \[\frac{(x+5)^{2}}{25} + \frac{(y-3)^{2}}{49} = 1\],we follow these principles to determine the ellipse's properties.

Ellipse Center Coordinates

The center of an ellipse is analogous to the heart of its geometry; it's where the major and minor axes intersect.
The coordinates of the center can be easily extracted by examining the terms \((x-h)\) and \((y-k)\)in the standard form of the ellipse equation.
In our specific equation, \(\frac{(x+5)^{2}}{25} + \frac{(y-3)^{2}}{49} = 1\),the center \((h,k)\) corresponds to the point where the changes in x and y would equal zero. Based on the signs in the equation, the center is at \((-5, 3)\). Recognizing and plotting the center is the first step to drawing the ellipse on a coordinate plane and provides a reference point for all other features of the ellipse.

Ellipse Vertices

The vertices of an ellipse are points of intersection where the ellipse touches the longest diameter, which is also known as the major axis.
For our given ellipse, the vertices are separated from the center \((-5, 3)\) by \(\pm b\) along the y-axis because this is where the semi-major axis lies. The value of \(b\) can be deduced from the equation; it is the square root of the larger denominator. In this case, \(\sqrt{49}=7\). Hence, adding and subtracting \(b\) from the y-coordinate of the center gives us the vertices at \((-5, 3\pm7)\), which calculates to \((-5, -4)\) and \((-5, 10)\). These are key points for sketching the full shape of the ellipse as they define the extent of the ellipse along its major axis.

Ellipse Foci

An ellipse's foci are two points located symmetrically on the major axis, and they are crucial in defining an ellipse's shape according to its definition involving distances. The foci (plural of focus) lie at a fixed distance 'c' from the center, where Â\(c^2 = b^2 - a^2\).To find these focal points for the ellipse in question, we first determine 'c' by calculating the difference between Â\(b^2\) and Â\(a^2\), which in this case is \(24\). Taking the square root gives us Â\(c ≈ 4.899\).
These foci profoundly influence the ellipse's curvature, as all points on the ellipse are equidistant from the foci. The coordinates of the foci are thus calculated by adding and subtracting 'c' from the y-coordinate of the center due to the vertical orientation of our ellipse. Rounding to three significant digits, the location of the foci for our given ellipse is at \((-5, 3\pm4.899)\), which gives the approximate coordinates \((-5, -1.899)\) and \((-5, 7.899)\). These points are not merely theoretical; they have practical applications such as in optics, where the properties of the foci explain the reflective characteristics of elliptical shapes.