Question

Discuss the graph of \(\left(\mathrm{x}^{2} / 25\right)+\left(\mathrm{y}^{2} / 9\right)=1\).

Short answer

The graph of the given ellipse \(\frac{x^2}{25} + \frac{y^2}{9} = 1\) has its center at the origin (0,0), with a semi-major axis of length 5 units along the x-axis and a semi-minor axis of length 3 units along the y-axis. The ellipse has a horizontal orientation.

Step-by-step solution

Identifying the center of the ellipse

The given equation represents an ellipse in standard form, which can be written as: \[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\] Comparing this standard equation to the given equation: \(\frac{x^2}{25} + \frac{y^2}{9} = 1\), we can see that the center of the ellipse is at point (h, k) = (0,0).

Finding the semi-major and semi-minor axes

In the standard equation of an ellipse, \(a^2\) and \(b^2\) are the squared lengths of the semi-major and semi-minor axes respectively. In our given equation, \(a^2 = 25\) and \(b^2 = 9\). Taking the square root of both these numbers, we get: Semi-major axis (a) = 5 Semi-minor axis (b) = 3

Determining the orientation of the ellipse

The orientation of the ellipse depends on whether the semi-major axis is on the x-axis or y-axis. Since the denominator of the x-term is larger (25 > 9), the semi-major axis is along the x-axis, and the semi-minor axis is along the y-axis.

Graphing the ellipse

To graph the ellipse, follow these steps: 1. Mark the center of the ellipse at the origin (0,0). 2. Draw the semi-major axis along the x-axis in both directions from the center, with a length equal to the value of 'a' (5 units). 3. Draw the semi-minor axis along the y-axis in both directions from the center, with a length equal to the value of 'b' (3 units). 4. Sketch the ellipse by creating a smooth curve that passes through the endpoints of both axes. The graph of the ellipse \(\frac{x^2}{25} + \frac{y^2}{9} = 1\) is completed. The ellipse has its center at the origin (0,0), a semi-major axis length of 5 units along the x-axis, and a semi-minor axis length of 3 units along the y-axis. The general orientation of the ellipse is horizontal.

Key concepts

Standard Form of an Ellipse

Understanding the standard form of an ellipse is crucial when attempting to graph it. The equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) represents an ellipse's standard form, where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively. If \(a > b\), the ellipse is stretched further along the x-axis, making it horizontally oriented. Otherwise, it's vertically oriented.

The equation given in the exercise, \(\frac{x^2}{25} + \frac{y^2}{9} = 1\), aligns perfectly with this standard form. Here, 'a' and 'b' are found by taking the square roots of the denominators in the equation. Understanding this will set the foundation for graphing the ellipse correctly.

Semi-Major and Semi-Minor Axes

An ellipse has two axes, called the semi-major and semi-minor axes. These are not just lengths but are also key to defining its shape. For the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(a\) is associated with the semi-major axis while \(b\) is associated with the semi-minor axis.

In the given exercise, \(a^2 = 25\) and \(b^2 = 9\), so when square-rooted, \(a = 5\) and \(b = 3\). These values tell us how far the ellipse stretches along the x and y axes. The semi-major axis impacts the width, and the semi-minor axis affects the height of the ellipse.

Ellipse Orientation

Orientation of an ellipse is determined by comparing the lengths of the semi-major and semi-minor axes. This characteristic informs us about the ellipse's alignment to the coordinate axes. When \(a > b\), the ellipse extends more along the x-axis, known as a horizontal ellipse, and if \(b > a\), it's a vertical ellipse.

In our exercise, the fact that \(a = 5\) is larger than \(b = 3\) indicates a horizontal orientation. This means the longest diameter of the ellipse runs left to right, parallel to the x-axis. Remembering this detail will help in accurately sketching the ellipse on a graph.

Ellipse Center

The center of an ellipse is analogous to the heart of it, where it's symmetrical on all sides. For the standard form \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), the center is located at the point \( (h, k) \). If the equation lacks an \(h\) or \(k\), as is the case in our equation \(\frac{x^2}{25} + \frac{y^2}{9} = 1\), the center defaults to \( (0, 0) \), the origin.

Identifying the center is vital as it is the starting point from which the axes are drawn, and it helps us to accurately place the ellipse within the coordinate plane. In the exercise, the origin is the center, which allowed us to graph the ellipse symmetrically around this point.