Lesson 10 Operating on a Shoestring Solidify Understanding

Ready

The rectangle in figure $B$ is a translation of the rectangle in figure $A$ . Write the equations of the two diagonals of rectangle $A B C D$ in point-slope form. Then write the equations of the two diagonals of $A^{'} B^{'} C^{'} D^{'}$ .

1.

2.

3.

The equations of the diagonals of rectangle $J K L M$ are $y_{1} = \frac{5}{8} x$ and $y_{2} = - \frac{5}{8} x$ .

Rectangle $J K L M$ is then translated so that its diagonals intersect at the point $(12, - 9)$ .

Write the equation of the diagonals of the translated rectangle.

Set

An ellipse is centered at the origin. A right triangle is drawn from $F_{2}$ on the $x$ -axis to the origin, $R$ , and then along $y$ -axis to point $S (0, b)$ .

4.

Find the lengths of the three sides of $△ F_{2} R S$ .

5.

Triangle $F_{2} R S$ is reflected across the $y$ -axis so that one of the vertices is now at $F_{1}$ . Find the sum of $F_{1} S + F_{2} S$ .

6.

Explain how the sum of $F_{1} S + F_{2} S$ connects to the definition of an ellipse.

For problems 7 and 8, use the given foci and length of the major axis to find the equation of the ellipse.

7.

Foci located at $(0, - 12)$ and $(0, 12)$

Major axis length $= 26$

8.

Foci located at $(- 4, 0)$ and $(4, 0)$

Major axis length $= 10$

For problems 9 and 10, use the graph of the ellipse with indicated foci to find the equation of the ellipse.

9.

10.

Not all ellipses are centered at the origin. An ellipse with center $(h, k)$ is translated $h$ units horizontally and $k$ units vertically. The standard form of the equation of an ellipse with center at $C (h, k)$ and whose vertices horizontally and vertically are $\pm a$ and $\pm b$ , respectively, is $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$ .

Write an equation, in standard form, for each ellipse based on the given center, $C$ , and the given values for the major axis, $a$ , and the minor axis, $b$ . Each of these ellipses has a horizontal major axis.

11.

$C (- 2, 3)$ , $a = 4$ , $b = 2$

12.

$C (5, 2)$ , $a = 5$ , $b = 3$

13.

$C (- 4, - 7)$ , $a = 10$ , $b = 8$

14.

$C (6, - 5)$ , $a = 7$ , $b = \sqrt{11}$

Write the equation of each ellipse in standard form. Identify the center. Then graph the ellipse.

15.

$4 x^{2} + y^{2} - 32 x - 4 y + 52 = 0$

Standard form:

Center:

16.

$16 x^{2} + 9 y^{2} - 96 x + 72 y + 144 = 0$

Standard form:

Center:

Go

Use the graph to find the missing values.

17.

$f (- 1) =$

18.

$g (0) =$

19.

$f (x) = g (x)$

$x =$

20.

$f (x) - g (x) = 0$

$x =$

21.

$f (x) \cdot g (x) = 0$

$x =$

22.

$f (2) + g (2) =$

23.

$f (0) - g (0) =$