Problem 70 - Precalculus with Limits

Question

In Exercises 69-72, sketch the graph of the ellipse, using latera recta (see Exercise 68). \(\dfrac{x^2}{4} + \dfrac{y^2}{1} = 1\)

Step-by-Step Solution

Answer

The ellipse is centred at the origin with vertices at (2, 0) and (-2, 0). The semi-major axis has a length of 4, while the semi-minor axis has a length of 2. The latera recta pass through the vertices and have length 1, each extending from y = 0.5 to y = -0.5.

Step 1: Identify the Semi-Major and Semi-Minor Axes

From the given equation, \(a^2\) is under the x term and \(a\) is therefore equal to \( \sqrt{4} = 2 \). \n Similarly, \(b^2\) is under the y term which implies that \(b = \sqrt{1} = 1\). The ellipse is therefore centred at the origin (0, 0) with semi-major axis along the x-axis of length 2 and semi-minor axis along the y-axis of length 1.

Step 2: Identify the Vertices

The vertices lie on the semi-major axis. Because the semi-major axis is along the x-axis and has length \(2a = 4 \), the vertices of the ellipse are at points \((2, 0) \) and \((-2, 0) \).

Step 3: Identify the Latera Recta

The latera recta of the ellipse have length \( \dfrac{2b^2}{a} = 1 \) and they are parallel to the minor axis. Therefore, they pass through the vertices and their end points lie on the minor axis. The length of the latera recta is the same on both sides of the centre, thus they lie on the line \(y = 0.5\) and \(y = -0.5\) respectively.

Step 4: Sketch the Graph

Now, the ellipse can be graphed accurately. Put the centre at the origin, and graph semi-major axis along the x-axis and semi-minor axis along the y-axis as per the identified lengths. Mark the vertices and correspondingly the ends of the latera recta. Then, smoothly draw curves that pass through these points to form an enclosed ellipse.

Key Concepts

Semi-Major and Semi-Minor Axes

Understanding the semi-major and semi-minor axes is fundamental when learning about ellipses. Think of an ellipse as a stretched circle. The longest line you can draw through the center of an ellipse is called the semi-major axis, and the shortest is the semi-minor axis.

In the given equation \(\dfrac{x^2}{4} + \dfrac{y^2}{1} = 1\), we identify the length of the semi-major axis as 2 units and the semi-minor axis as 1 unit. This information provides us with guidelines on how broadly or narrowly the ellipse is stretched out.

These axes not only define the shape but also help in identifying other crucial features of an ellipse, such as its vertices and foci, which are aligned along these axes. If you imagine the axes as guide rails, they make sure the shape of your ellipse stays consistent and proportional.

Vertices of an Ellipse

The vertices of an ellipse are points of intersection where the ellipse touches its longest diameter—the semi-major axis. In our exercise, with the semi-major axis value of 2 from the center in both directions along the x-axis, we establish our vertices' coordinates at \( (2, 0) \) and \( (-2, 0) \).

These points are valuable markers for drawing an accurate representation of the ellipse. They are the 'tips' of the ellipse and are essential when sketching since they establish the widest boundary of the shape. By pinpointing the vertices, you create a visual boundary that the curve of the ellipse will adhere to, ensuring that the sketch remains true to the mathematical properties of the shape.

Latera Recta

The concept of latera recta might not be as commonly known as axes and vertices, but it is just as significant in the study of ellipses. The term 'latera recta' refers to the line segments that are parallel to the semi-minor axis and pass through the ellipse's foci. They represent chords of the ellipse that are perpendicular to the major axis.

In our solution, the length of a latera recta is discovered using the formula \( \dfrac{2b^2}{a} = 1 \). Knowing this measurement helps to identify two additional points through which the ellipse will pass. These segments act like crossbeams that provide more structural details to the sketch of the ellipse—enhancing the accuracy with which the curve is drawn.

Conic Sections

Ellipses belong to the family of shapes known as conic sections. These shapes are formed by the intersection of a plane with a cone. Depending on the angle of intersection, you could get a circle, an ellipse, a parabola, or a hyperbola.

Our focus, the ellipse, is formed when the plane cuts through the cone at an angle that is oblique to the base of the cone but does not intersect the base. It's a fascinating concept because it means ellipses, much like the one defined by \(\dfrac{x^2}{4} + \dfrac{y^2}{1} = 1\), are not just flat sketches on paper—they are the result of slicing through a three-dimensional object.

The study of conic sections interlinks much of geometry and algebra, demonstrating how through different perspectives, one can encounter a diversity of forms and structures.