MAT039 Lesson 2: Lines and Circles

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Questions and Answers

What is the distance between the points (3, −2) and (0, 2)?

4 units

10 units

5 units (correct)

6 units

What is the formula to calculate the slope of a line through two points, P(x1, y1) and Q(x2, y2)?

$m_L = \frac{y2 - y1}{x2 - x1}$ (correct)

$m_L = \frac{y2 + y1}{x2 + x1}$

$m_L = \frac{y2 - y1}{x1 - x2}$

$m_L = \frac{y1 - y2}{x2 - x1}$

What is the angle of inclination of a vertical line?

$\pi$ radians

$\frac{\pi}{4}$ radians

0 radians

$\frac{\pi}{2}$ radians (correct)

If the angle of inclination of a non-horizontal line is 0 radians, what can be inferred about the line?

The line is horizontal. Signup and view all the answers

Which of the following statements is true about the distance formula between two points P and Q?

The formula is $d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$. Signup and view all the answers

What is the midpoint of the line segment joining the points (2, 3) and (4, 7)?

(3, 5) Signup and view all the answers

What is the value of the slope of a horizontal line?

0 Signup and view all the answers

For two points P(1, 2) and Q(3, 6), what is the angle of inclination of the line connecting them?

$\frac{\pi}{3}$ radians Signup and view all the answers

What is the y-coordinate of the midpoint M of the line segment joining the points (2, 3) and (4, 1)?

2 Signup and view all the answers

What is the slope of the line through the points (0, 0) and the midpoint M (3, 2)?

3 Signup and view all the answers

Which of the following is the equivalent equation of the line through (0, 0) with slope 3?

2x - 3y = 0 Signup and view all the answers

What is the y-coordinate of the intersection point of the lines 2x + 3y + 1 = 0 and x + 2y - 1 = 0?

3 Signup and view all the answers

Which equation represents the line L passing through the intersection point (-5, 3) and the point (0, -3)?

6x + 5y + 15 = 0 Signup and view all the answers

Under what condition are two lines L and l considered parallel?

m_L = m_l Signup and view all the answers

When are two lines L and l considered to be perpendicular?

m_L = -m_l Signup and view all the answers

If line L has a slope of 2, what would be the slope of a line perpendicular to it?

-2 Signup and view all the answers

What are the coordinates of point Q if P is (5, -4) and the midpoint of PQ is (-2, -1)?

(-9, 2) Signup and view all the answers

What is the slope of the line L with an angle of inclination of 3π/4 radians?

-1 Signup and view all the answers

What is the slope of the line passing through the points (-3, 4) and (2, -6)?

-2 Signup and view all the answers

Which of the following formulas represents the point-slope form of a line?

y - y1 = m(x - x1) Signup and view all the answers

What is the slope-intercept form of a line?

y = mx + b Signup and view all the answers

What can be said about the x-intercept and y-intercept of the line represented by the equation 7x + 3y - 6 = 0?

The slope is -7/3 and the y-intercept is (0, 2). Signup and view all the answers

To find the midpoint of a line segment joining the points (2, 3) and (4, 1), what will be the coordinates of the midpoint?

(3, 2) Signup and view all the answers

If the line has an x-intercept of -3 and a y-intercept of 4, which of the following equations describes this line in intercepts form?

x/(-3) + y/4 = 1 Signup and view all the answers

What does the distance from a point to a line represent?

The length of the perpendicular segment from the point to the line. Signup and view all the answers

Which equation gives the distance from a point P to the line Ax + By + C = 0?

d = rac{Ax_0 + By_0 - C}{ ext{sqrt}(A^2 + B^2)} Signup and view all the answers

If a line is given by the equation 3x - 7y + 2 = 0, what is the slope of the line?

rac{3}{7} Signup and view all the answers

For a line that is perpendicular to another with an average slope of 2, what is the slope of the perpendicular line?

-rac{1}{2} Signup and view all the answers

What is the distance from the point (5, -3) to the line given by 2x + 3y - 1 = 0?

0 Signup and view all the answers

What form should be used to find the equation of a line given a point and a slope?

Point-slope form Signup and view all the answers

How many points on the x-axis are at a unit distance from the line 3x + 4y + 5 = 0?

Two Signup and view all the answers

When a line has the equation 3x - 7y + 2 = 0 and is parallel to another line, what can be said about their slopes?

They are equal. Signup and view all the answers

What are the coordinates of point P if the midpoint of line segment PQ is at (3, 4) and the abscissa of P is 4?

(4, 6) Signup and view all the answers

How can you demonstrate that the quadrilateral with vertices A(1, 4), B(1, -3), C(-6, 4), and D(-6, -3) is a square?

By proving all angles are 90 degrees Signup and view all the answers

What is the distance from the point (1, 2) to the midpoint of the segment whose endpoints are (-1, -3) and (-3, -5)?

$rac{5 ext{ units}}{2}$ Signup and view all the answers

What value of a ensures the distance from (-1, 2) to the line given by the equation $ax - y + 9 = 0$ is 5 units?

4 Signup and view all the answers

What is the shortest distance between the two parallel lines described by the equations $2x - y - 2 = 0$ and $2x - y - 7 = 0$?

$5 ext{ units}$ Signup and view all the answers

What is the slope of the line represented by the equation $2x + 5y + 1 = 0$?

$-rac{2}{5}$ Signup and view all the answers

What is the equation of the line that has an angle of inclination of $\frac{5 ext{π}}{6}$ radians and a y-intercept of -4?

y = -\frac{\sqrt{3}}{3} x - 4 Signup and view all the answers

What is the equation of a line that is perpendicular to $x + y + 2 = 0$ and passes through the midpoint of the intercepts of this line?

y = -x + 1 Signup and view all the answers

Study Notes

Lines

Distance Definition: The undirected distance between two points (P(x_1, y_1)) and (Q(x_2, y_2)) is defined as the length of the line segment connecting them.
Distance Formula: Given two points in the plane, the distance (d) is calculated as: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
Midpoint Formula: The midpoint (M) of the line segment joining points (P) and (Q) is: [ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) ]
Angle of Inclination: The angle (\alpha) formed by a non-horizontal line with the positive x-axis ranges from 0 to (\pi) radians, with vertical lines having an angle of (\frac{\pi}{2}) radians, and horizontal lines having an angle of 0 radians.
Slope Definition: For a non-vertical line with angle of inclination (\alpha), the slope (m_L) is given by: [ m_L = \tan(\alpha) ]
Slope Between Two Points: For points (P(x_1, y_1)) and (Q(x_2, y_2)) where (x_1 \neq x_2), the slope is calculated as: [ m_L = \frac{y_2 - y_1}{x_2 - x_1} ]

Example Calculations

Distance Example: For points ((3, -2)) and ((0, 2)), [ d = \sqrt{(0 - 3)^2 + (2 - (-2))^2} = \sqrt{9 + 16} = 5 \text{ units} ]
Midpoint Example: Given midpoint ((-2, -1)) and point (P(5, -4)), find (Q(x_2, y_2)) using: [ -2 = \frac{5 + x_2}{2}, \quad -1 = \frac{-4 + y_2}{2} \implies x_2 = -9, y_2 = 2 \implies Q(-9, 2) ]
Slope from Angle: A line with angle of inclination (\frac{3\pi}{4}) has slope: [ m_L = \tan\left(\frac{3\pi}{4}\right) = -1 ]
Slope between points: For points ((-3, 4)) and ((2, -6)), [ m_L = \frac{-6 - 4}{2 - (-3)} = \frac{-10}{5} = -2 ]

Equation of a Line

Point-Slope Form: Given a point ((x_1, y_1)) and slope (m): [ y - y_1 = m(x - x_1) ]
Slope-Intercept Form: Given slope (m) and y-intercept (b): [ y = mx + b ]
Two-Point Form: For points ((x_1, y_1)) and ((x_2, y_2)): [ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) ]
Intercepts Form: Given x-intercept (a) and y-intercept (b): [ \frac{x}{a} + \frac{y}{b} = 1 ]

Parallel and Perpendicular Lines

Parallel Lines: Two non-vertical lines (L) and (l) are parallel if: [ m_L = m_l ]
Perpendicular Lines: Two non-vertical lines (L) and (l) are perpendicular if: [ m_L = -\frac{1}{m_l} ]

Distance from a Point to a Line

Distance Formula: The shortest distance (d) from point (P(x_0, y_0)) to line (Ax + By + C = 0) is: [ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} ]

Example Problems

Parallel Line Example: Line through point ((-4, -1)) parallel to line (3x - 7y + 2 = 0): [ \text{slope } = \frac{3}{7} \implies y + 1 = \frac{3}{7}(x + 4) ]
Perpendicular Line Example: Line through ((5, -3)) perpendicular to line through points ((-3, -1)) and ((1, 5)): [ \text{slope of } l = \frac{6}{4} = \frac{3}{2} \implies m_L = -\frac{2}{3} ]

Exercises

Midpoint Calculation: Find points (P) and (Q) knowing their midpoint.
Distance Calculation: Find the distance from a point to a line, and the shortest distance between two parallel lines.
Slope and Intercept: Determine the slope and y-intercepts from various line equations.

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Description

This quiz covers the basics of lines, focusing on the distance formula between two points in the plane. It examines definitions and theorems that are essential for understanding geometrical concepts involving lines and circles. Test your knowledge and understanding of these foundational principles in mathematics.