Problem 50 - Thomas Calculus

Question

Exercises \(49-52\) give equations for ellipses and tell how many units up or down and to the right or left each ellipse is to be shifted. Find an equation for the new ellipse, and find the new foci, vertices, and center. $$ \frac{x^{2}}{2}+y^{2}=1, \quad \text { right } 3, \text { up } 4 $$

Step-by-Step Solution

Answer

New equation: \(\frac{(x-3)^{2}}{2} + (y-4)^{2} = 1\), Center: (3, 4), Vertices: \((3\pm\sqrt{2}, 4)\), Foci: \((3\pm 1, 4)\)."

Step 1: Identify the original equation format

The original equation is \(\frac{x^{2}}{2} + y^{2} = 1\). This equation is almost in the standard form for an ellipse, \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), but with \(h = 0\) and \(k = 0\). Here, \(a^{2} = 2\) and \(b^{2} = 1\).

Step 2: Express the shift of the ellipse

The problem tells you the ellipse is to be moved 3 units to the right and 4 units up. Thus, \(h = 3\) and \(k = 4\) for the shifted ellipse.

Step 3: Write the new equation of the ellipse

Substitute \(h = 3\), \(k = 4\), \(a^{2} = 2\), and \(b^{2} = 1\) into the standard ellipse equation: \[\frac{(x-3)^{2}}{2} + \frac{(y-4)^{2}}{1} = 1\].

Step 4: Determine the new center of the ellipse

The new center of the ellipse is at the shifted position, which is \((h, k) = (3, 4)\).

Step 5: Calculate the new vertices

The vertices of the ellipse are located along the major and minor axes. Since \(a^{2} = 2\), \(a = \sqrt{2}\), and \(y\) has \(b = 1\). The major axis is horizontal, so vertices are at \((3\pm\sqrt{2}, 4)\).

Step 6: Find the foci

The foci of an ellipse can be found using \(c = \sqrt{a^{2} - b^{2}}\). Here, \(c = \sqrt{2 - 1} = 1\). Thus, the foci are located at \((3\pm 1, 4)\).

Key Concepts

Equation of an ellipse

An ellipse is an elongated circle and its equation in standard form looks like this:

• \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)

This equation helps us understand the shape and position of the ellipse on a graph.
In this form, \(h\) and \(k\) are the coordinates of the center of the ellipse.
The values \(a\) and \(b\) decide how wide or tall the ellipse is.For the given exercise, our original equation is \(\frac{x^2}{2} + y^2 = 1\).
Here, \(h = 0\) and \(k = 0\) mean the ellipse is centered at the origin.
With \(a^2 = 2\) and \(b^2 = 1\), it tells us that the ellipse stretches more horizontally than vertically.
When calculating \(a\) and \(b\), we take the square root: \(a = \sqrt{2}\) and \(b = 1\). This tells us how far the ellipse extends from its center along the axes.

Shift of ellipse

To shift an ellipse means to move it from one position to another on the coordinate plane.
It's like sliding the entire shape across the graph.In our exercise, the ellipse is moved 3 units right and 4 units up.
This is described mathematically by changing \(h\) and \(k\) in the standard elliptical equation:

• \((h, k) = (3, 4)\)

This moves the center of the ellipse from the origin \((0, 0)\) to \((3, 4)\).
Now the new equation becomes \(\frac{(x-3)^2}{2} + \frac{(y-4)^2}{1} = 1\).
The shift doesn't change the shape of the ellipse, only its position.
This is important because all the internal distances measured in the ellipse, like \(a\) and \(b\), stay the same.

Ellipse foci and vertices

Ellipses have special points called foci and vertices, which tell us a lot about their geometry.

Vertices

Vertices are located at the widest points of the ellipse, on the major and minor axes.
The major axis is the longer one. For our ellipse, given \(a = \sqrt{2}\), vertices are found by adding and subtracting \(a\) from \(x\)-coordinate of the center.
Vertices are at \((3 \pm \sqrt{2}, 4)\). This means two vertices are symmetrically placed on the horizontal axis of our shifted ellipse.

Foci

Foci are two critical points inside the ellipse that help define its shape.
They are calculated using: \(c = \sqrt{a^2 - b^2}\).
In this exercise, \(c = 1\). So the foci are at \((3 \pm 1, 4)\), axially aligned like the vertices but just a bit closer to the center.
The vertices and foci provide a complete understanding of both the size and the orientation of the ellipse on the coordinate plane.