Question: top-left: Multiple Choice
top-left: Identify the choice that best completes the statement or answers the question. MUST SHOW ALL WORK FOR CREDIT
top-left: 1. Find the distance $d(P_1, P_2)$ between the points $P_1$
top-left: and $P_2$.
top-left: $P_1 = (-1, -1); P_2 = (-1, 2)$
top-left: a. 3
top-left: b. 2
top-left: c. $\sqrt{3}$
top-left: d. 4
top-left: 2. Find the midpoint of the line segment joining
top-left: the points $P_1$ and $P_2$.
top-left: $P_1 = (7, 1); P_2 = (-16, -16)$
top-left: a. $(9, 15)$
top-left: b. $(-9/2, -15/2)$
top-left: c. $(23/2, 17/2)$
top-left: d. $(-9, -15)$
top-left: 3. Find the slope-intercept form of the equation:
top-left: Horizontal: containing the point $(-7/8, 4)$
top-left: a. $y = 4$
top-left: b. $y = -4$
top-left: c. $y = -7/8$
top-left: d. $y = 0$
top-left: 4. Find the slope of the line containing the two
top-left: points.
top-left: $(-9, -6); (-9, -7)$
top-left: a. $-1$
top-left: b. $1$
top-left: c. undefined
top-left: d. $0$
top-right: 5. Find an equation for the line:
top-right: Containing the points $(7, 0)$ and $(0, -6)$;
top-right: a. $y = -6/7 x - 6$
top-right: b. $y = -6/7 x + 7$
top-right: c. $6x + 7y = 42$
top-right: d. $6x - 7y = 42$
top-right: 6. Write the standard form of the circle with
top-right: radius $r$ and center $(h, k)$.
top-right: $r = \sqrt{2}$, $(h, k) = (-4, 4)$
top-right: a. $(x - 4)^2 + (y + 4)^2 = 2$
top-right: b. $(x + 4)^2 + (y - 4)^2 = 2$
top-right: c. $(x - 4)^2 + (y + 4)^2 = 4$
top-right: d. $(x + 4)^2 + (y - 4)^2 = 4$
top-right: 7. Find the center $(h, k)$ and radius $r$ of the circle
top-right: with the given equation.
top-right: $x^2 + y^2 = 16$
top-right: a. $(h, k) = (4, 4); r = 4$
top-right: b. $(h, k) = (0, 0); r = 4$
top-right: c. $(h, k) = (0, 0); r = 16$
top-right: d. $(h, k) = (4, 4); r = 16$
bottom-right: 8. Find an equation for the line:
bottom-right: Parallel to the line $y = -3x$; containing the point $(8, 6)$
bottom-right: a. $y = -3x$
bottom-right: b. $y - 6 = -3x - 8$
bottom-right: c. $y = -3x - 30$
bottom-right: d. $y = -3x + 30$
bottom-right: 9. Find an equation:
bottom-right: Perpendicular to the line $y = \frac{1}{7} x + 6$; containing
top-right: the point $(4, -3)$
bottom-right: a. $y = -7x - 25$
bottom-right: b. $y = -\frac{1}{7} x - \frac{25}{7}$
bottom-right: c. $y = -7x + 25$
bottom-right: d. $y = 7x - 25$
1. **Find the distance $d(P_1, P_2)$ between $P_1=(-1,-1)$ and $P_2=(-1,2)$**
The distance formula is:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Calculate:
$$d = \sqrt{(-1 - (-1))^2 + (2 - (-1))^2} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3$$
Answer: a. 3
2. **Find the midpoint of the segment joining $P_1=(7,1)$ and $P_2=(-16,-16)$**
Midpoint formula:
$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$
Calculate:
$$M = \left( \frac{7 + (-16)}{2}, \frac{1 + (-16)}{2} \right) = \left( \frac{-9}{2}, \frac{-15}{2} \right)$$
Answer: b. $\left(-\frac{9}{2}, -\frac{15}{2}\right)$
3. **Find slope-intercept form of horizontal line through $(-\frac{7}{8}, 4)$**
A horizontal line has equation:
$$y = k$$
where $k$ is the y-coordinate of the point.
Answer: a. $y = 4$
4. **Find slope of line through $(-9,-6)$ and $(-9,-7)$**
Slope formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Calculate denominator:
$$x_2 - x_1 = -9 - (-9) = 0$$
Slope is undefined because division by zero.
Answer: c. undefined
5. **Find equation of line through $(7,0)$ and $(0,-6)$**
Slope:
$$m = \frac{-6 - 0}{0 - 7} = \frac{-6}{-7} = \frac{6}{7}$$
Use point-slope form with point $(0,-6)$:
$$y - (-6) = \frac{6}{7}(x - 0)$$
$$y + 6 = \frac{6}{7}x$$
$$y = \frac{6}{7}x - 6$$
Rewrite in standard form:
$$7y = 6x - 42$$
$$6x - 7y = 42$$
Answer: d. $6x - 7y = 42$
6. **Write standard form of circle with radius $r=\sqrt{2}$ and center $(-4,4)$**
Standard form:
$$(x - h)^2 + (y - k)^2 = r^2$$
Calculate $r^2$:
$$r^2 = (\sqrt{2})^2 = 2$$
Plug in values:
$$(x - (-4))^2 + (y - 4)^2 = 2$$
$$(x + 4)^2 + (y - 4)^2 = 2$$
Answer: b. $(x + 4)^2 + (y - 4)^2 = 2$
7. **Find center and radius of circle $x^2 + y^2 = 16$**
Rewrite as:
$$(x - 0)^2 + (y - 0)^2 = 16$$
Center: $(0,0)$
Radius:
$$r = \sqrt{16} = 4$$
Answer: b. $(0,0); r=4$
8. **Find equation of line parallel to $y = -3x$ through $(8,6)$**
Parallel lines have same slope $m = -3$.
Use point-slope form:
$$y - 6 = -3(x - 8)$$
$$y - 6 = -3x + 24$$
$$y = -3x + 30$$
Answer: d. $y = -3x + 30$
9. **Find equation perpendicular to $y = \frac{1}{7}x + 6$ through $(4,-3)$**
Slope of given line:
$$m = \frac{1}{7}$$
Slope of perpendicular line:
$$m_{\perp} = -7$$
Use point-slope form:
$$y - (-3) = -7(x - 4)$$
$$y + 3 = -7x + 28$$
$$y = -7x + 25$$
Answer: c. $y = -7x + 25$