Q 11.46. ... [FREE SOLUTION]

Chapter 11: Q 11.46. (page 1111)

Short Answer

Expert verified

The graph of an ellipse is shown below:

Step by step solution

Step 1. Given information

The given ellipse is $\frac{{(x - 1)}^{2}}{25} + \frac{{(y + 3)}^{2}}{16} = 1$ .

Step 2. Write the equation in standard form.

The standard form of an ellipse with center $(h, k)$ is given by $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1 \dots (1)$ .

Compare the given equation with standard form and conclude that $\frac{{(x - 1)}^{2}}{25} + \frac{{(y + 3)}^{2}}{16} = 1$ is in standard form and the center is $(1, - 3)$ .

Step 3. Find whether the major axis is horizontal or vertical.

Compare the equation $(1)$ with $\frac{{(x - 1)}^{2}}{25} + \frac{{(y + 3)}^{2}}{16} = 1$ to find $a^{2}$ and $b^{2}$ .

$\begin{array}{rcl} a^{2} & = & 25 \\ b^{2} & = & 16 \end{array}$

Since $25 > 16$ and $25$ is in the $x^{2}$ term, the major axis will be horizontal.

Step 4. Find the endpoints of the major axis and the minor axis.

The value is $a^{2} = 25$ .

This implies that $a = \pm 5$ .

This follows the distance from the center to the vertices is $5$ .

The value is $b^{2} = 16$ .

This implies that $b = 4$ .

This follows that the minor axis is vertical and the distance from the center to the endpoints of the minor axis is $4$ .

Step 5. Draw the ellipse.

The graph of an ellipse is shown below:

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Write the equation in standard form.

Step 3. Find whether the major axis is horizontal or vertical.

Step 4. Find the endpoints of the major axis and the minor axis.

Step 5. Draw the ellipse.

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