Problem 42 - Thomas Calculus

Question

The ellipse \(\left(x^{2} / 9\right)+\left(y^{2} / 25\right)=1\) is shifted 3 units to the left and 2 units down to generate the ellipse $$\frac{(x+3)^{2}}{9}+\frac{(y+2)^{2}}{25}=1$$ a. Find the foci, vertices, and center of the new ellipse. b. Plot the new foci, vertices, and center, and sketch in the new ellipse.

Step-by-Step Solution

Answer

The center is at (-3, -2), vertices at (-3, 3) and (-3, -7), foci at (-3, 2) and (-3, -6).

Step 1: Understanding the problem

We are given an equation of an ellipse and informed that it has been shifted. The original ellipse equation is \(\frac{x^2}{9} + \frac{y^2}{25} = 1\). It is moved 3 units left and 2 units down to \(\frac{(x+3)^2}{9} + \frac{(y+2)^2}{25} = 1\). We need to find the foci, vertices, and center of this new ellipse.

Step 2: Determine the center of the new ellipse

The center of an ellipse in standard form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) is at \((h, k)\). From the given equation \(\frac{(x+3)^2}{9} + \frac{(y+2)^2}{25} = 1\), the center is at \((-3, -2)\).

Step 3: Identify the vertices of the new ellipse

For an ellipse of the form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) with \(a^2 < b^2\), the vertices are located at \((h, k\pm b)\). Here, \(b^2 = 25\), so \(b = 5\). Hence, the vertices are at \((-3, -2\pm 5) \Rightarrow (-3, 3)\) and \((-3, -7)\).

Step 4: Find the foci of the new ellipse

The foci of an ellipse are determined by the formula \(c^2 = b^2 - a^2\), where \(b^2 = 25\) and \(a^2 = 9\). Thus, \(c^2 = 25 - 9 = 16\), meaning \(c = 4\). The foci are at \((h, k\pm c)\), leading to \((-3, -2\pm 4)\). Thus, the foci are \((-3, 2)\) and \((-3, -6)\).

Step 5: Plot and sketch the new ellipse

Plot the center at \((-3, -2)\). Mark the vertices at \((-3, 3)\) and \((-3, -7)\). Mark the foci at \((-3, 2)\) and \((-3, -6)\). Sketch the ellipse, ensuring it is elongated vertically since \(b > a\).

Key Concepts

Ellipse Properties

Ellipses are fascinating geometric shapes that belong to a group called *conic sections*. Even though they might look like simple stretched circles, they possess unique properties that set them apart. An ellipse has two axes, a major and a minor one. The longest one, known as the major axis, determines the longest distance across the ellipse, while the shortest one, called the minor axis, defines the width of the ellipse.

Every ellipse has two special points called foci. The sum of the distances from any point on the ellipse to these foci is always constant, which lends the ellipse its distinct shape. The center of the ellipse is midway between the foci and serves as a point of symmetry. For our shifted ellipse equation, \((x+3)^{2}/9 + (y+2)^{2}/25 = 1\), the center is \(-3, -2\).

The vertices of an ellipse are the points where the ellipse intersects its major axis. These provide significant information regarding the size and orientation of the ellipse. Knowing the center and vertices helps in visualizing and sketching the ellipse more accurately.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is all about using coordinates to define geometric shapes. In the case of ellipses, equations come to play a crucial role in describing their position on a graph.

The standard form of an ellipse's equation is \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where \(h, k\) represents the center of the ellipse. By shifting an ellipse, as seen in our exercise, we translate its position on the coordinate plane without altering its shape.

• To shift an ellipse horizontally to the left by 3 units: replace \(x\) with \(x + 3\)
• To shift it vertically down by 2 units: replace \(y\) with \(y + 2\)

Through this coordinate transformation, we see how basic mathematics can manipulate and control the orientation of these shapes in space.

Conic Sections

Conic sections are shapes created as a plane intersects a cone. These include circles, ellipses, parabolas, and hyperbolas, each having unique properties and equations that define their structure.

An ellipse, like the one in our exercise, is formed when the plane cuts through both nappes of a cone but not parallel to its base. This forms a closed curve with an oval-like shape.

Conic sections share equations that reveal their geometry:

• Each conic section's equation can be derived from its general quadratic form.
• For ellipses, identifying the coefficients allows us to find their orientation, shape, center, and dimensions.

The ability to identify an ellipse within these equation sets highlights the profound connection between algebra and geometry, revealing how abstract numbers paint detailed pictures of shapes in the world of mathematics.