Find Center, Vertices, Eccentricity, Foci, Latus Rectum, And Directrices Of The Ellipse X² + 2y² = 10

In this comprehensive guide, we will delve into the ellipse defined by the equation x² + 2y² = 10. Our goal is to identify and calculate several key features of this ellipse, including its center, vertices, eccentricity, foci, length of the latus rectum, and the equations of its directrices. This exploration will provide a thorough understanding of the ellipse's geometry and its position within the Cartesian coordinate system. We will break down each step with detailed explanations and calculations, ensuring clarity and accuracy. Understanding these properties is crucial for various applications in physics, engineering, and mathematics, making this analysis highly valuable for students and professionals alike.

To begin, we need to convert the given equation into the standard form of an ellipse equation. The standard form allows us to easily identify the semi-major and semi-minor axes, which are crucial for determining other properties. The given equation is:

x² + 2y² = 10

To convert this to standard form, we divide both sides by 10:

(x²/10) + (2y²/10) = 1

Simplifying the second term, we get:

(x²/10) + (y²/5) = 1

This is the standard form of the ellipse equation, which can be written as:

(x²/a²) + (y²/b²) = 1

where a² = 10 and b² = 5. This form immediately tells us important information about the ellipse's shape and orientation. By recognizing the standard form, we can easily compare our equation to the general form and extract the values of a and b, which are essential for further calculations. The standard form not only simplifies the process of finding these values but also provides a clear visual representation of the ellipse's dimensions and its placement on the coordinate plane.

The center of the ellipse is the midpoint of both the major and minor axes. In the standard form equation (x²/a²) + (y²/b²) = 1, if there are no shifts in the x and y terms (i.e., no terms like (x - h)² or (y - k)²), the center of the ellipse is at the origin (0, 0). In our case, the equation is:

(x²/10) + (y²/5) = 1

Since there are no shifts in x or y, the center of this ellipse is at (0, 0). This means the ellipse is symmetrically positioned around the origin, making it easier to analyze its other properties. The center serves as a reference point for all other calculations, such as determining the vertices, foci, and directrices. Knowing the center is the first step in fully understanding the ellipse's geometry, as it provides the central anchor point from which all other features are measured and calculated. The simplicity of identifying the center in this case underscores the importance of having the equation in standard form, as it allows for quick recognition of key characteristics.

The vertices of an ellipse are the endpoints of the major axis. The major axis is the longer axis of the ellipse. In our equation, (x²/10) + (y²/5) = 1, we have a² = 10 and b² = 5. Since a² > b², the major axis is along the x-axis. Therefore, a = √10, and the vertices are located at (±a, 0).

The vertices are thus at (±√10, 0), which are approximately (±3.16, 0). These points represent the farthest horizontal extent of the ellipse from its center. Understanding the vertices is critical because they define the overall length and orientation of the ellipse. The major axis, which connects these vertices, is the ellipse's longest diameter and plays a crucial role in determining its shape. By pinpointing the vertices, we gain a clearer picture of the ellipse's dimensions and its position relative to the coordinate axes. This information is vital for various applications, including graphical representation and problem-solving in geometry and related fields.

The eccentricity (e) of an ellipse is a measure of how much the ellipse deviates from a perfect circle. It is defined as:

e = √(1 - (b²/a²))

In our case, a² = 10 and b² = 5. Plugging these values into the formula, we get:

e = √(1 - (5/10)) = √(1 - 0.5) = √0.5 = √(1/2) = 1/√2

Thus, the eccentricity e = 1/√2, which is approximately 0.707. The eccentricity value ranges between 0 and 1, where 0 represents a perfect circle and 1 represents a parabola. An eccentricity of 1/√2 indicates that the ellipse is moderately elongated. The higher the eccentricity, the more stretched the ellipse appears. This value provides important information about the shape of the ellipse and helps in visualizing its geometry. Eccentricity is a key parameter in various scientific applications, such as describing the orbits of planets and satellites. A precise calculation of eccentricity allows for accurate modeling and prediction of elliptical paths in diverse physical systems.

The foci of an ellipse are two points on the major axis that are crucial for defining the shape of the ellipse. The distance from the center to each focus is given by c, where:

c = √(a² - b²)

In our equation, a² = 10 and b² = 5. Plugging these values into the formula, we get:

c = √(10 - 5) = √5

Since the major axis is along the x-axis, the foci are located at (±c, 0). Therefore, the foci are at (±√5, 0), which are approximately (±2.24, 0). The foci are essential for understanding the reflective properties of the ellipse. For example, any ray emanating from one focus will reflect off the ellipse and pass through the other focus. This property has significant applications in optics and acoustics. The location of the foci helps define the overall shape and dimensions of the ellipse, providing a key element in its geometric description. Accurate determination of the foci is important for both theoretical understanding and practical applications of ellipses.

The latus rectum of an ellipse is a line segment passing through a focus, perpendicular to the major axis, with endpoints on the ellipse. The length of the latus rectum is given by the formula:

Length of Latus Rectum = (2b²)/a

In our case, a = √10 and b² = 5. Plugging these values into the formula, we get:

Length of Latus Rectum = (2 * 5) / √10 = 10 / √10 = √10

Thus, the length of the latus rectum is √10, which is approximately 3.16. The latus rectum provides a measure of the