Question

Find the equation of the ellipse defined by the given information. Sketch the ellipse. Foci: (2,±2)\(;\) vertices: (2,±7)

Short answer

The equation is \( \frac{(x-2)^2}{45} + \frac{y^2}{49} = 1 \).

Step-by-step solution

Identify the center of the ellipse

The foci and vertices both have a constant x-coordinate of 2. This means the center of the ellipse is located at (2,0). The vertices are at (2,±7), thus indicating the center is midpoint between them.

Determine the semi-major axis

The distance between the center (2,0) and each vertex, (2,7) or (2,-7), is 7. Therefore, the semi-major axis \(a\) is 7. This is the distance from the center to a vertex along the vertical axis.

Determine the semi-minor axis

The distance from the center (2,0) to a focus (2,±2) is 2. Using the relationship \(c^2 = a^2 - b^2\) where \(c\) is the distance from the center to a focus, we substitute \(a = 7\) and \(c = 2\) to find \(b\). \[ c^2 = a^2 - b^2 \]\[ 2^2 = 7^2 - b^2 \]\[ 4 = 49 - b^2 \]\[ b^2 = 45 \]\[ b = \sqrt{45} \].

Write the equation of the ellipse

Substituting \(a\), \(b\), and the center (h,k) into the standard equation of an ellipse \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \), we get:\[ \frac{(x-2)^2}{45} + \frac{y^2}{49} = 1 \].

Sketch the ellipse

To sketch the ellipse, plot the center at (2,0). The vertices will be at (2,7) and (2,-7) since the semi-major axis length is 7, vertically aligned. The foci are within the ellipse at (2,2) and (2,-2). With \(b = \sqrt{45}\), the semi-minor axis extends approximately 6.7 units horizontally from the center, creating ellipse boundaries at about (2+\sqrt{45}, 0) and (2-\sqrt{45}, 0).

Key concepts

Foci of an Ellipse

The foci of an ellipse are two special points located along its major axis. These points are significant because the sum of the distances from any point on the ellipse to each focus is constant.
In the given problem, the foci are located at \((2, \pm 2)\). This tells us that the major axis is vertical, as evidenced by the changing y-coordinates while the x-coordinate remains constant at 2.

• Each focus is symmetric with respect to the center of the ellipse, which also lies on the major axis.
• The distance from the center of the ellipse to each focus is represented as \(c\).

In this case, since the center is at \( (2,0) \), the distance \( c \) is \( 2 \). The parameter \( c \) is essential to determining the relation among the other parameters of the ellipse.

Vertices of an Ellipse

Vertices of an ellipse are the endpoints of the major axis. They are critical points that define the overall length and orientation of the ellipse.
For this exercise, the vertices provided are \((2, \pm 7)\). This highlights that the major axis is again vertical because the vertices change with the y-coordinate while keeping the x-coordinate constant at 2.

• The center of the ellipse sits at the midpoint between the vertices. Therefore, the center here is \((2,0)\).
• The distance from the center to any vertex is called the semi-major axis, denoted by \(a\).

Since the vertices are at \(y = 7\) and \(y = -7\), the semi-major axis, \(a\), is \(7\). The length of the major axis is thus \(2a = 14\).

Equation of an Ellipse

The equation of an ellipse depends on its orientation and the lengths of its axes. The standard form of the equation for an ellipse centered at \((h,k)\) with vertical major axis is:\[\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\]
The exercise gave us the center of the ellipse as \((2,0)\), with \(a = 7\) (semi-major axis) and \(b = \sqrt{45}\) (semi-minor axis).
Plugging these values in, we derive the equation:\[\frac{(x-2)^2}{45} + \frac{y^2}{49} = 1\]

• This formula reflects the shape and dimensions of the ellipse by relating the coordinates
• The equation tells us how the x and y variables should behave to draw the path of the ellipse.

With this understanding, we can construct the graph of the ellipse accurately.

Semi-Major and Semi-Minor Axes

The semi-major and semi-minor axes are the principal dimensions that describe an ellipse. The semi-major axis represents the longest radius, extending from the center to a vertex, while the semi-minor axis is the shortest radius extending orthogonally from the center.
In this context, the semi-major axis \(a\) is \(7\) because it stretches from the center \((2,0)\) to any vertex \((2, \pm7)\).
The semi-minor axis \(b\) is calculated using the relation between \(a\), \(b\), and \(c\):\[ c^2 = a^2 - b^2\]Given \( c = 2 \) and \( a = 7 \), solving the equation yields \(b = \sqrt{45}\).
This measurement is key to constructing an accurate ellipse, indicating that the ellipse is wider when measured vertically compared to horizontally.