MTH120 College Algebra Chapter 8.1

Analytic geometry, also known as coordinate geometry, is a branch of mathematics that combines geometry and algebra. It involves studying geometric figures and solving geometric problems using a coordinate system. In analytic geometry, points, lines, curves, and shapes are represented using algebraic equations and coordinates.

Key concepts and elements of analytic geometry include:

1. Coordinate Systems: The most common coordinate systems are the Cartesian coordinate system and the polar coordinate system. In the Cartesian system, points are represented by ordered pairs (x, y), while in the polar system, points are represented by an angle θ and a distance r from the origin.

2. Equations of Lines: Lines in the Cartesian coordinate system are described by linear equations, such as y = mx + b (slope-intercept form) or Ax + By = C (standard form). The slope of a line is given by m, and (x, y) satisfies the equation of the line.

3. Equations of Circles: The equation of a circle with its center at (h, k) and radius r is given by (x - h)^2 + (y - k)^2 = r^2. This equation allows the representation of circles in the Cartesian plane.

4. Conic Sections: Conic sections include circles, ellipses, parabolas, and hyperbolas. They can be described using algebraic equations, and the eccentricity of each conic section determines its shape.

5. Transformations: Transformations, such as translation, rotation, reflection, and scaling, are used to modify geometric shapes. These transformations are represented using matrix algebra and allow for the study of symmetry and similarity.

6. Distance Formula: The distance between two points (x1, y1) and (x2, y2) in the Cartesian plane is calculated using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).

7. Midpoint Formula: The midpoint between two points (x1, y1) and (x2, y2) is calculated using the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).

8. Analyzing Curves: Analytic geometry can be used to study curves and their properties. For instance, the calculus-based technique of finding derivatives and integrals can be applied to analyze the slope, concavity, and area under curves.

9. Vectors: Vectors are quantities that have both magnitude and direction and are often used to describe lines, forces, and motion. Vectors can be represented in two or three dimensions and manipulated using vector algebra.

Analytic geometry is widely used in various fields, including physics, engineering, computer graphics, and computer-aided design (CAD), where it provides a powerful tool for modeling and solving complex geometric problems.

8.1 The Ellipse

An ellipse is a geometric shape that resembles a flattened circle. It can be defined as the set of all points in a plane such that the sum of the distances from two fixed points (called the foci) is constant. The midpoint between the foci is called the center of the ellipse. The line segment passing through the center and connecting two opposite points on the ellipse is the major axis, while the line segment perpendicular to the major axis and also passing through the center is the minor axis.

Key features and concepts related to ellipses include:

1. Foci (Focus Singular): The foci are two fixed points inside the ellipse. The distance between the foci and the center of the ellipse is a constant value, denoted as 2c. The major axis is the segment that passes through the foci.

2. Center: The center is the midpoint between the two foci. It is also the point where the major and minor axes intersect.

3. Major Axis: The major axis is the longer of the two axes and passes through the foci and the center.

4. Minor Axis: The minor axis is the shorter of the two axes, and it is perpendicular to the major axis, passing through the center.

5. Semi-Major and Semi-Minor Axes: The semi-major axis (a) is half the length of the major axis, while the semi-minor axis (b) is half the length of the minor axis. These values determine the size and shape of the ellipse.

6. Eccentricity (e): The eccentricity of an ellipse is a measure of how elongated or flattened it is. It is defined as the ratio of the distance from a point on the ellipse to one of the foci (c) to the semi-major axis (a). Mathematically, $�=�\mathrm{/}�$.

7. Equation of an Ellipse: The standard equation of an ellipse with its center at the origin (0,0) and major axis along the x-axis is: $\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$

   If the major axis is not along the x-axis, the equation can be transformed to accommodate the rotation. The general equation of an ellipse with its center at (h, k) and major and minor axes lengths a and b is: $\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

8. Eccentricity and Shape: The value of eccentricity determines the shape of the ellipse. When $0<�<1$, the ellipse is elongated (more like a circle when e is close to 0), while when $�=1$, it becomes a parabola, and when $�>1$, it becomes a hyperbola.

Ellipses are encountered in various fields of science and engineering, from astronomy (planetary orbits) to optics (shape of lenses and mirrors) and engineering (design of gear teeth). Understanding the properties and equations of ellipses is important for solving real-world problems involving these geometric shapes.

To write the equation of an ellipse in standard form, you need to determine its key characteristics: the coordinates of the center, the lengths of the major and minor axes, and the orientation (whether it is aligned with the x-axis or y-axis). The standard form of the equation of an ellipse depends on these factors. Here are the general steps to write the equation of an ellipse in standard form:

1. Identify the Center: Determine the coordinates of the center of the ellipse. The center is denoted as (h, k).

2. Determine the Major and Minor Axes: Find the lengths of the major and minor axes. The major axis has length 2a, and the minor axis has length 2b.

3. Determine the Orientation: Determine whether the major axis is aligned with the x-axis or the y-axis. If it is aligned with the x-axis, the equation will be in the form $\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$. If it is aligned with the y-axis, the equation will be in the form $\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$.

4. Calculate the Values of a and b: The values of a and b are the lengths of the semi-major and semi-minor axes, respectively. They can be calculated from the given information about the major and minor axes.

5. Write the Equation: Based on the orientation, center, and values of a and b, write the equation in standard form. Here are the standard forms for both orientations:

   • If the major axis is aligned with the x-axis (horizontal orientation): $\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

   • If the major axis is aligned with the y-axis (vertical orientation): $\frac{(�-�{)}^2}{{�}^2}+\frac{(�-ℎ{)}^2}{{�}^2}=1$

In these equations, (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.

Here's an example of how to write the equation of an ellipse in standard form:

Example: Suppose you have an ellipse with a center at (2, 3), a major axis of length 8, and a minor axis of length 6, aligned with the x-axis. To write the equation in standard form, follow these steps:

1. Center: (h, k) = (2, 3)
2. Major Axis (2a): 8, so a = 4
3. Minor Axis (2b): 6, so b = 3
4. Orientation: Aligned with the x-axis

Now, write the equation in standard form: $\frac{(�-2{)}^2}{4^2}+\frac{(�-3{)}^2}{3^2}=1$

This is the equation of the ellipse in standard form.

To derive the equation of an ellipse centered at the origin, you can start with the general equation of an ellipse in standard form:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$.

In this equation, $�$ is the semi-major axis (the longer radius) and $�$ is the semi-minor axis (the shorter radius) of the ellipse. If the ellipse is centered at the origin, it means that the center of the ellipse is at the point (0, 0).

The equation for the ellipse centered at the origin is based on the Pythagorean theorem. The distance from the center (0, 0) to any point on the ellipse ($�,�$) is equal to the sum of the distances along the x-axis and y-axis. Using the distance formula, this can be expressed as:

$\sqrt{{�}^2+{�}^2}$.

Since the ellipse is centered at the origin, this distance is equal to the semi-major axis $�$. Therefore, you can write:

$\sqrt{{�}^2+{�}^2}=�$.

Now, to eliminate the square root and simplify the equation, square both sides:

${�}^2+{�}^2={�}^2$.

This is the equation of an ellipse centered at the origin. However, it's not in the standard form I mentioned earlier, which includes the semi-minor axis $�$. To get the standard form, you need to consider the relationship between $�$ and (b.

The relationship between the semi-major axis ($�$) and the semi-minor axis ($�$) is determined by the eccentricity ($�$) of the ellipse. The eccentricity is a measure of how "elongated" the ellipse is and is defined as:

$�=\sqrt{1-\frac{{�}^2}{{�}^2}}$.

You can rearrange this equation to solve for ${�}^2$:

${�}^2={�}^2-{�}^2{�}^2$.

Now, substitute this expression for ${�}^2$ back into the equation ${�}^2+{�}^2={�}^2$:

${�}^2+{�}^2={�}^2-{�}^2{�}^2$.

Divide both sides by ${�}^2$ to obtain the standard form of the equation for an ellipse centered at the origin:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2-{�}^2{�}^2}=1$.

So, the standard equation of an ellipse centered at the origin is:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2(1-{�}^2)}=1$.

This equation describes an ellipse with its center at the origin, where $�$ is the semi-major axis and $�$ is the eccentricity.

Let's go through an example to derive the equation of an ellipse centered at the origin and put it in standard form.

Example: Suppose you have an ellipse with a semi-major axis of $�=4$ units and a semi-minor axis of $�=3$ units. First, we need to find the eccentricity ($�$) using the formula:

$�=\sqrt{1-\frac{{�}^2}{{�}^2}}$.

In this case, $�=4$ and $�=3$, so:

$�=\sqrt{1-\frac{3^2}{4^2}}=\sqrt{1-\frac{9}{16}}=\sqrt{\frac{16}{16}-\frac{9}{16}}=\sqrt{\frac{7}{16}}=\frac{\sqrt{7}}{4}$.

Now, we can write the equation for the ellipse centered at the origin. The standard form is:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$.

Plugging in the values of $�$ and $�$ from our example:

$\frac{{�}^2}{4^2}+\frac{{�}^2}{3^2}=1$.

Now, you can simplify this equation:

$\frac{{�}^2}{16}+\frac{{�}^2}{9}=1$.

This is the equation of the ellipse centered at the origin with a semi-major axis of 4 units and a semi-minor axis of 3 units. The eccentricity $�$ for this ellipse is (\frac{\sqrt{7}}{4}).

In this equation, you can see that the semi-major axis ($�$) and semi-minor axis ($�$) are used to determine the shape of the ellipse, while the eccentricity ($�$) describes how "elongated" the ellipse is.

To write the equation of an ellipse centered at the origin in standard form, you need to determine the lengths of the semi-major axis ($�$) and the semi-minor axis ($�$), as well as the direction (whether it's aligned with the x-axis or y-axis), and then use this information to form the equation. The standard form for an ellipse centered at the origin is:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$

Here are a few examples to illustrate how to write these equations:

Example 1: An ellipse centered at the origin with the semi-major axis along the x-axis and the semi-minor axis along the y-axis.

Let's say the semi-major axis ($�$) is 5 units, and the semi-minor axis ($�$) is 3 units. The equation in standard form will be:

$\frac{{�}^2}{5^2}+\frac{{�}^2}{3^2}=1$

This represents an ellipse centered at the origin, stretched in the x-direction by a factor of 5 and in the y-direction by a factor of 3.

Example 2: An ellipse centered at the origin with the semi-major axis along the y-axis and the semi-minor axis along the x-axis.

If the semi-major axis ($�$) is 4 units, and the semi-minor axis ($�$) is 2 units, the equation in standard form will be:

$\frac{{�}^2}{2^2}+\frac{{�}^2}{4^2}=1$

This represents an ellipse centered at the origin, stretched in the y-direction by a factor of 4 and in the x-direction by a factor of 2.

Example 3: An ellipse centered at the origin with equal semi-major and semi-minor axes.

If the semi-major axis ($�$) and the semi-minor axis ($�$) are equal, say 3 units each, the equation in standard form will be:

$\frac{{�}^2}{3^2}+\frac{{�}^2}{3^2}=1$

This represents a circle centered at the origin with a radius of 3 units.

In each of these examples, the standard form of the equation represents an ellipse centered at the origin. The values of $�$ and $�$ determine the size and shape of the ellipse, and whether it's stretched more in the x-direction or the y-direction.

The standard form of the equation for an ellipse centered at the origin, which is also known as the general form, is:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$.

In this equation:

• $�$ represents the length of the semi-major axis (the longer radius).
• $�$ represents the length of the semi-minor axis (the shorter radius).

The values of $�$ and $�$ determine the size and shape of the ellipse. If $�>�$, the ellipse is stretched more in the x-direction, and if $�>�$, it's stretched more in the y-direction.

Here are some variations of the standard form for different types of ellipses:

1. Circle (Special Case): When $�=�$, the equation represents a circle centered at the origin. In this case, the equation simplifies to:

   $\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$,

   where $�$ is the radius of the circle.

2. Horizontal Ellipse: When $�>�$, the ellipse is stretched more in the x-direction, and the equation is:

   $\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$.

3. Vertical Ellipse: When $�>�$, the ellipse is stretched more in the y-direction, and the equation becomes:

   $\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$.

In all cases, the center of the ellipse is at the origin (0,0), and the values of $�$ and $�$ determine the size and shape of the ellipse.

To write the equation of an ellipse in standard form when given the coordinates of its vertices and foci, you need to determine the lengths of the semi-major axis ($�$), the semi-minor axis ($�$), and the orientation (whether it's aligned with the x-axis or y-axis). Here's how you can do it:

1. Find the Length of the Major Axis ($2�$): The distance between the vertices of the ellipse is equal to the length of the major axis ($2�$). If you're given the coordinates of the vertices, you can use the distance formula to find $2�$. The formula is:

   $2�=\sqrt{({�}_1-{�}_2{)}^2+({�}_1-{�}_2{)}^2},$

   where $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ are the coordinates of two vertices.

2. Find the Length of the Minor Axis ($2�$): The distance between the foci of the ellipse is equal to the length of the major axis ($2�$). If you're given the coordinates of the foci, you can use the distance formula to find $2�$. The formula is the same as above, but with the coordinates of the foci.

3. Determine the Orientation: The orientation of the ellipse (whether it's aligned with the x-axis or y-axis) depends on the arrangement of the vertices and foci. If the foci lie along the x-axis, the major axis is horizontal; if the foci lie along the y-axis, the major axis is vertical.

Once you've determined the values of $2�$ and $2�$ and the orientation of the ellipse, you can write the equation in standard form.

If it's a horizontal ellipse (major axis along the x-axis):

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1.$

If it's a vertical ellipse (major axis along the y-axis):

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1.$

In these equations, $�$ represents the length of the semi-major axis and $�$ represents the length of the semi-minor axis.

To write the equation of an ellipse that is not centered at the origin, you need to account for both the position of the center and the lengths of the semi-major and semi-minor axes. The standard form for an ellipse centered at $(ℎ,�)$ with the semi-major axis $�$ and the semi-minor axis $�$ aligned with the x and y-axes respectively is:

$\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$.

Here are a few examples to illustrate how to write these equations:

Example 1: Write the equation of an ellipse centered at $(2,-3)$ with a semi-major axis of 4 units and a semi-minor axis of 3 units.

The equation in standard form is:

$\frac{(�-2{)}^2}{4^2}+\frac{(�+3{)}^2}{3^2}=1$.

This represents an ellipse centered at $(2,-3)$, stretched more in the x-direction by a factor of 4 and in the y-direction by a factor of 3.

Example 2: Write the equation of an ellipse centered at $(-1,2)$ with the semi-major axis of 5 units along the y-axis and a semi-minor axis of 3 units along the x-axis.

The equation in standard form is:

$\frac{(�+1{)}^2}{3^2}+\frac{(�-2{)}^2}{5^2}=1$.

This represents an ellipse centered at $(-1,2)$, stretched more in the y-direction by a factor of 5 and in the x-direction by a factor of 3.

Example 3: Write the equation of an ellipse centered at $(3,4)$ with equal semi-major and semi-minor axes of 2 units each.

The equation in standard form is:

$\frac{(�-3{)}^2}{2^2}+\frac{(�-4{)}^2}{2^2}=1$.

This represents a circle centered at $(3,4)$ with a radius of 2 units.

In each of these examples, the values of $ℎ$ and $�$ determine the center of the ellipse, while the values of $�$ and $�$ determine the size and shape of the ellipse.

The standard form of the equation for an ellipse not centered at the origin, with the center at $(ℎ,�)$, is given by:

$\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

In this equation:

• $(ℎ,�)$ are the coordinates of the center of the ellipse.
• $�$ is the length of the semi-major axis (the longer radius).
• $�$ is the length of the semi-minor axis (the shorter radius).

The values of $�$ and $�$ determine the size and shape of the ellipse, and the coordinates $(ℎ,�)$ represent the center.

Here are a few variations based on the orientation of the ellipse:

1. Horizontal Ellipse: When the major axis is aligned with the x-axis, the equation is:

   $\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

2. Vertical Ellipse: When the major axis is aligned with the y-axis, the equation is:

   $\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

In these equations:

• If $�>�$, the ellipse is stretched more along the x-direction.
• If $�>�$, the ellipse is stretched more along the y-direction.

Example:

Consider an ellipse with its center at $(3,-1)$, a semi-major axis of length 4, and a semi-minor axis of length 3. The equation in standard form would be:

$\frac{(�-3{)}^2}{4^2}+\frac{(�+1{)}^2}{3^2}=1$

This represents an ellipse centered at $(3,-1)$, stretched more in the x-direction by a factor of 4 and in the y-direction by a factor of 3.

To write the equation of an ellipse not centered at the origin when given the coordinates of its vertices and foci, you'll need to follow these steps:

1. Find the Center ($ℎ,�$): The center of the ellipse is the midpoint between the vertices. If $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ are the coordinates of the vertices, the center $(ℎ,�)$ is given by:

   $ℎ=\frac{{�}_1+{�}_2}{2},�=\frac{{�}_1+{�}_2}{2}$

2. Find the Length of the Major Axis ($2�$): The distance between the vertices is equal to the length of the major axis. Use the distance formula:

   $2�=\sqrt{({�}_1-{�}_2{)}^2+({�}_1-{�}_2{)}^2}$

3. Find the Distance Between Foci ($2�$): The distance between the foci is equal to $2�$, where $�$ is the distance from the center to each focus. Use the distance formula with the coordinates of the foci.

4. Find $�$: Use the relationship between $�$, $�$, and $�$ in an ellipse:

   $�=\sqrt{{�}^2-{�}^2}$

   Solving for $�$:

   $�=\sqrt{{�}^2-{�}^2}$

5. Determine Orientation: Check the orientation based on the arrangement of the vertices and foci. If the foci lie along the x-axis, the major axis is horizontal. If the foci lie along the y-axis, the major axis is vertical.

Now, you can write the equation in standard form based on the center $(ℎ,�)$, the lengths of the semi-major axis ($�$) and semi-minor axis ($�$), and the orientation.

If it's a horizontal ellipse (major axis along the x-axis):

$\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

If it's a vertical ellipse (major axis along the y-axis):

$\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

To graph an ellipse centered at the origin, you can follow these steps:

1. Determine the semi-major and semi-minor axes:

   • The semi-major axis ($�$) is the longer radius, and it determines how far the ellipse extends along its major axis.
   • The semi-minor axis ($�$) is the shorter radius, and it determines how far the ellipse extends along its minor axis.

2. Determine the orientation:

   • If the ellipse is elongated more horizontally (wider than it is tall), the major axis is along the x-axis, and the semi-major axis ($�$) is along the x-direction. In this case, $�>�$.
   • If the ellipse is elongated more vertically (taller than it is wide), the major axis is along the y-axis, and the semi-major axis ($�$) is along the y-direction. In this case, $�>�$.

3. Sketch the ellipse:

   • Start at the origin (0,0) and draw the major axis in the direction determined by the orientation.
   • Mark the endpoints of the major axis, which will be $�$ units away from the origin.
   • Mark the endpoints of the minor axis, which will be $�$ units away from the origin.
   • Sketch the ellipse by connecting the points on the major and minor axes, maintaining the same shape and proportions.

4. Add labels:

   • Label the lengths of the semi-major and semi-minor axes, $�$ and $�$, respectively.
   • You can also add the coordinates of the endpoints of the major and minor axes to provide additional information.

Here are two examples to illustrate how to graph ellipses centered at the origin:

Example 1: Graphing a horizontal ellipse:

Suppose you have an ellipse with a semi-major axis ($�$) of 4 units and a semi-minor axis ($�$) of 2 units. Since $�>�$, it's a horizontal ellipse. Here's how you can graph it:

• Start at the origin (0,0).
• Move 4 units to the right along the x-axis and mark a point.
• Move 4 units to the left along the x-axis and mark another point.
• Move 2 units upward and downward along the y-axis from the origin, marking two more points.
• Connect the points to form the ellipse.

Example 2: Graphing a vertical ellipse:

Suppose you have an ellipse with a semi-major axis ($�$) of 3 units and a semi-minor axis ($�$) of 5 units. Since $�>�$, it's a vertical ellipse. Here's how you can graph it:

• Start at the origin (0,0).
• Move 5 units upward along the y-axis and mark a point.
• Move 5 units downward along the y-axis and mark another point.
• Move 3 units to the right and left along the x-axis from the origin, marking two more points.
• Connect the points to form the ellipse.

Remember to label the lengths of the axes and any other relevant information on your graph.

Graphing an ellipse not centered at the origin involves a few additional steps compared to graphing an ellipse centered at the origin. Here are the steps to graph an ellipse not centered at the origin, along with examples:

Step 1: Determine the Center (h, k)

• The center of the ellipse is given by (h, k). Determine these coordinates based on the problem or equation you have.

Step 2: Determine the Semi-Major and Semi-Minor Axes (a and b)

• The semi-major axis (a) and semi-minor axis (b) determine the size and shape of the ellipse. If you are given these values, that's great. Otherwise, you can find them from the equation or additional information.

Step 3: Determine the Orientation

• Depending on the arrangement of the center and foci, you can determine whether the major axis is aligned with the x-axis or the y-axis.

Step 4: Graph the Ellipse

• Use the information gathered in the previous steps to draw the ellipse:
  • Start at the center (h, k).
  • Determine the endpoints of the major axis, which will be 'a' units away from the center in the direction of the major axis.
  • Determine the endpoints of the minor axis, which will be 'b' units away from the center in the direction of the minor axis.
  • Sketch the ellipse by connecting these points.
  • Label the center, lengths of the semi-major and semi-minor axes, and any other relevant information.

Let's go through a couple of examples:

Example 1: Graph the ellipse with the equation $\frac{(�-2{)}^2}{4}+\frac{(�+3{)}^2}{9}=1$.

In this case:

• Center: (2, -3)
• Semi-Major Axis ($�$): 3 (since $\sqrt{9}=3$)
• Semi-Minor Axis ($�$): 2 (since $\sqrt{4}=2$)
• Major axis is along the y-axis.

1. Start at the center (2, -3).
2. Move 3 units up and down along the y-axis from the center and mark points.
3. Move 2 units left and right along the x-axis from the center and mark points.
4. Connect the points to sketch the ellipse.

Example 2: Graph the ellipse with center (1, -2), semi-major axis of 5 units, and semi-minor axis of 3 units.

In this case:

• Center: (1, -2)
• Semi-Major Axis ($�$): 5
• Semi-Minor Axis ($�$): 3
• Major axis is along the x-axis.

1. Start at the center (1, -2).
2. Move 5 units to the left and right along the x-axis from the center and mark points.
3. Move 3 units up and down along the y-axis from the center and mark points.
4. Connect the points to sketch the ellipse.

Remember to label the center, lengths of the axes, and any other relevant information on your graph.

Solving applied problems involving ellipses often requires a good understanding of the properties and equations of ellipses. Here are some examples of applied problems involving ellipses and how to solve them:

Example 1: Satellite Orbits

A satellite is in an elliptical orbit around the Earth with the following information:

• The semi-major axis ($�$) is 10,000 kilometers.
• The semi-minor axis ($�$) is 8,000 kilometers.
• The Earth's center is at one focus of the ellipse.

Find the distance of the satellite from the Earth's center and the closest and farthest points from the Earth.

Solution:

1. To find the distance of the satellite from the Earth's center, you can use the formula for the distance of a point on an ellipse from the center:

   $�=\sqrt{{�}^2+{�}^2}$

   In this case, $(�,�)$ is the center of the ellipse (0, 0), and the distances are $�=10,000$ km and $�=8,000$ km. So:

   $$$$

2. To find the closest and farthest points from the Earth, you can use the distances $�$ and $�$. The closest point is the center of the ellipse, which is 6,000 km from the Earth's center. The farthest point is the end of the semi-major axis, which is 10,000 km from the Earth's center.

Example 2: Solar Concentrators

A solar concentrator uses an elliptical mirror to focus sunlight onto a solar panel. The mirror has a semi-major axis of 2 meters and a semi-minor axis of 1 meter.

Find the focal points of the mirror and determine where the solar panel should be placed to receive the focused sunlight.

Solution:

1. To find the focal points, you can use the relationship between the semi-major axis ($�$), the semi-minor axis ($�$), and the distance to the foci ($�$) in an ellipse:

   $�=\sqrt{{�}^2-{�}^2}=\sqrt{2^2-1^2}=\sqrt{3}\text{ meters}$

   The focal points are located at $(\pm �,0)$, which is $(\pm \sqrt{3},0)$ meters.

2. To determine where the solar panel should be placed, it should be positioned at one of the focal points, where the sunlight is focused. So, the solar panel should be placed at $(\sqrt{3},0)$ meters.

These are just a couple of examples of how to solve applied problems involving ellipses. Depending on the specific problem, you may need to use different properties of ellipses and adapt the approach accordingly.

To write the equation of an ellipse in standard form and identify the end points of the major and minor axes as well as the foci, you'll need to complete the square for both $�$ and $�$ terms. The standard form of the equation for an ellipse is:

$\frac{(�-ℎ{)}^2}{{�}^2}+\frac{(�-�{)}^2}{{�}^2}=1$

where $(ℎ,�)$ is the center, $�$ is the semi-major axis, and $�$ is the semi-minor axis.

For all problems:

1. Group ${�}^2$ terms and $�$ terms together, and complete the square.
2. Group ${�}^2$ terms and $�$ terms together, and complete the square.
3. Move constant terms to the other side of the equation.
4. Rewrite the equation in standard form.

Now, let's apply these steps to each problem:

22. $4{�}^2+22�+14{�}^2-124�+227=0$

Completing the square for $�$:

$4({�}^2+5.5�)+14{�}^2-124�+227=0$ $4({�}^2+5.5�+\frac{5.5^2}{4})+14{�}^2-124�+227-4\cdot \frac{5.5^2}{4}=0$

Completing the square for $�$:

$4({�}^2+5.5�+\frac{5.5^2}{4})+14({�}^2-9�)+227-4\cdot \frac{5.5^2}{4}-14\cdot \frac{9^2}{14}=0$

Simplify:

$4(�+\frac{5.5}{2}{)}^2+14(�-\frac{9}{2}{)}^2=100$

Divide both sides by 100:

$\frac{(�+\frac{5.5}{2}{)}^2}{25}+\frac{(�-\frac{9}{2}{)}^2}{\frac{100}{14}}=1$

This is the standard form. Now, identify $ℎ$, $�$, $�$, $�$, and then find the endpoints of the major and minor axes and the foci.

1. $4{�}^2+20�+21{�}^2-110�+101=0$

2. ${�}^2+2�+150{�}^2-1301�+2301=0$

3. $4{�}^2+24�+25{�}^2+200�+336=0$

You can follow similar steps for the other problems. Remember to factor out the coefficient of ${�}^2$ and ${�}^2$ before completing the square, and then simplify.