[Math 20] Lec 1.5 Lines and Circles

Summary

The video provides an in-depth exploration of the equations of lines and circles. It covers essential concepts like the slope of a line, point-slope form, slope-intercept form, and the general equation of a line. The video also discusses perpendicular and parallel lines, finding equations of lines passing through specific points, and the properties of tangent lines to circles. The explanation includes formulas, step-by-step examples, and a focus on how to find the slope and intercept of lines, as well as the equation of a circle and its tangent.

Takeaways

• 📝 The video explains the equations of lines and circles, starting with a review of a unique line and its slope.
• 📏 The slope of a line is determined by the formula: slope (m) = (y2 - y1) / (x2 - x1), representing the steepness of the line.
• 🔄 To find the equation of a line, use the point-slope form: y - y1 = m(x - x1), which is crucial for describing lines.
• 🔗 The slope-intercept form of a line is y = mx + b, where b is the y-intercept of the line.
• 🔍 Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.
• 🔄 The video demonstrates how to find the equation of a line that passes through two points or is perpendicular to another line.
• ⚪ The equation of a circle is given as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
• 📏 The distance formula between two points is used to relate points on a circle to its center.
• 🧮 The general form of a circle’s equation can be expanded to find specific values for the center and radius.
• 📐 Tangent lines to a circle intersect it at exactly one point and are perpendicular to the radius at the point of tangency.

Q & A

What is the slope of a line and how is it determined?

The slope of a line represents its steepness and is determined using the formula: slope (m) = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

How do you find the equation of a line using a point and the slope?

The equation of a line can be found using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

What is the slope-intercept form of a line equation?

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept, the point where the line crosses the y-axis.

How do you find the equation of a line that passes through two points?

To find the equation of a line passing through two points, first calculate the slope using m = (y2 - y1) / (x2 - x1), then use the point-slope form y - y1 = m(x - x1) with one of the points.

What does it mean for two lines to be parallel?

Two lines are parallel if they have the same slope, meaning the slopes of the two lines are equal (m1 = m2).

What is the relationship between the slopes of two perpendicular lines?

Two lines are perpendicular if the product of their slopes is -1. This means if one line has a slope of m, the other line has a slope of -1/m.

How can you find the equation of a line perpendicular to another line and passing through a given point?

First, find the slope of the given line and take the negative reciprocal to get the slope of the perpendicular line. Then, use the point-slope form with the given point to find the equation.

What is the definition of a circle in a plane?

A circle is defined as a set of points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is the radius.

How do you write the equation of a circle given its center and radius?

The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².

What is a tangent line to a circle, and how is it characterized?

A tangent line to a circle is a line that touches the circle at exactly one point. At the point of tangency, the line is perpendicular to the radius drawn to the point of tangency.

Outlines

00:00

📐 Introduction to Equations of Lines and Circles

The video begins by introducing the concept of equations related to lines and circles. It recalls the importance of the slope in determining the steepness of a line. The formula for calculating the slope using two points on a line (x1, y1) and (x2, y2) is provided, along with a detailed explanation of how to derive the equation of a line using the point-slope form.

05:10

✏️ Deriving the Equation of a Line

This section explains how to find the equation of a line that passes through two points. The video demonstrates using the slope formula, substituting into the point-slope form, and finally converting it into the slope-intercept form. An example is given, showing how to derive the equation of a line using specific points and the negative reciprocal of the slope.

10:14

🛤️ Perpendicular Lines and Equations

The discussion shifts to lines that are perpendicular to each other, explaining that two lines are perpendicular if the product of their slopes is -1. The process to find the equation of a line that is perpendicular to another given line is outlined, using a specific example to illustrate the method.

15:15

⭕ Introduction to Circles and Their Equations

This section defines what a circle is and introduces the equation of a circle in the plane. The center-radius form of the circle's equation is presented, along with a brief explanation of how to derive it using the distance formula between two points. The general form of the circle's equation is also introduced, providing a method to determine the center and radius.

20:33

📏 Finding the Equation of a Circle

Here, the video walks through the process of finding the equation of a circle given its center and a point on the circle. It involves substituting the given center and point into the center-radius form and solving for the radius. The general equation of the circle is then provided, showing how to relate it to the center-radius form.

25:39

🔗 Tangent Lines to Circles

The final section explores tangent lines to circles, which intersect the circle at exactly one point. The video explains how to find the equation of a tangent line to a given circle at a specific point, using the perpendicularity property of the tangent and the radius. An example is provided to demonstrate this method.

📚 Conclusion on Lines and Circles

The video concludes with a summary of the key concepts discussed, including equations of lines, circles, and tangent lines. The instructor hopes that the viewers have understood the material, and thanks them for watching.

Keywords

💡Slope

The slope of a line refers to its steepness, represented by the ratio of the vertical change to the horizontal change between two points on the line. In the video, the slope is discussed as a fundamental concept for determining the equation of a line. The formula for calculating the slope, m = (y2 - y1) / (x2 - x1), is highlighted, emphasizing its importance in defining the characteristics of a line.

💡Point-Slope Form

The point-slope form is an equation of a line that uses the slope and a specific point on the line to express the line's equation: y - y1 = m(x - x1). This form is central to understanding how lines are defined in the video, providing a method to describe a line when the slope and a point on the line are known.

💡Slope-Intercept Form

The slope-intercept form of a line's equation is y = mx + b, where m is the slope and b is the y-intercept. This form is particularly useful for quickly identifying the slope and where the line crosses the y-axis. The video explains how to derive this form from the point-slope form, underlining its practical use in finding the equation of a line.

💡Parallel Lines

Parallel lines are lines in the same plane that never intersect because they have the same slope. In the video, the concept of parallel lines is discussed to show how lines with equal slopes behave, reinforcing the idea that their slopes must be identical.

💡Perpendicular Lines

Perpendicular lines intersect at a 90-degree angle, and the product of their slopes is -1. The video uses this concept to illustrate how to identify and work with lines that are perpendicular, especially in the context of determining the equation of a line given its relationship to another line.

💡Circle

A circle is defined as the set of all points in a plane that are equidistant from a fixed point, known as the center. The video introduces the equation of a circle as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius, highlighting its application in geometry.

💡Center of a Circle

The center of a circle is the fixed point from which all points on the circle are equidistant. In the video, the center is denoted by (h, k) and is crucial for forming the equation of the circle. Understanding the center's role is key to working with circle equations.

💡Radius

The radius of a circle is the distance from the center to any point on the circle. The video discusses how the radius, denoted as r, is used in the circle's equation and how it determines the size of the circle. The concept is vital for solving problems related to circles.

💡Tangent Line

A tangent line to a circle is a line that touches the circle at exactly one point. The video explains how to find the equation of a tangent line to a circle, emphasizing its geometric properties, particularly its perpendicularity to the radius at the point of tangency.

💡General Form of a Circle

The general form of a circle's equation is expressed as Ax² + Ay² + Bx + Cy + D = 0. The video covers how to derive this form from the standard form, and explains how it can be used to find the circle's center and radius through algebraic manipulation.

Highlights

Introduction to equations of lines and circles, emphasizing the importance of understanding the slope and determining the uniqueness of a line.

Explanation of slope as the steepness of a line, and how to calculate it using the formula (y2 - y1) / (x2 - x1).

Introduction of the point-slope form equation: y - y₁ = m(x - x₁), highlighting its importance in describing a line given a point and its slope.

Derivation of the slope-intercept form equation: y = mx + b, and its significance in finding the equation of a line with a known y-intercept.

Discussion of how to find the equation of a line that passes through two points using the slope-intercept form.

Explanation of parallel and perpendicular lines, including the conditions for parallel lines (same slope) and perpendicular lines (product of slopes equals -1).

Detailed steps to find the equation of a line perpendicular to a given line and passing through a specific point using the negative reciprocal of the slope.

Introduction to circles as a set of points equidistant from a center point, and the equation of a circle: (x - h)² + (y - k)² = r².

Application of the distance formula to derive the equation of a circle with a known center and radius.

Explanation of the general form of a circle's equation and how to convert it to the standard form to find the center and radius.

Worked example on finding the equation of a circle given a center and a point on the circle, demonstrating the use of the standard form.

Introduction to tangent lines to circles, defining a tangent line as one that intersects a circle at exactly one point.

Illustration of the special property of tangent lines: the radius at the point of tangency is perpendicular to the tangent line.

Example problem on finding the equation of a tangent line to a circle at a specific point, including deriving the slope and using the slope-intercept form.

Conclusion summarizing the key concepts of lines, circles, and their interactions, reinforcing the importance of the equations and methods discussed.

Transcripts

hello so in this video we're going to

talk about the equations of

lines and circles

so let us recall come on

unique line so determine

is a unique length

on the line and the slope of the line

so pakistan's slope of the line that is

the steepness

of airline so back at nothing

steepness of a line so let's say this is

your point

when we say steepness or the slope

uh or because say some point maraming

lines

um so parama specify nothing

we have to determine the slope of the

line

and to get the slope of a line so let us

recall so let's say this is your line

you have two points

x to y two and x one y y1

you can compute for the slope m using

the formula y2 minus y1 all over

x2 minus x1 or the change in y divided

by the change in

x and how do we get the equation of a

line so he's having an attitude

and at a point in the slope so let's say

the slope of her line is

m and x not y

naught is a point on the line so pan

another major describe it online at all

so let's say we have a generic point on

your line so

how are these x and y related

so let me tell you a formula in a slope

so sabina

and this that the slope is equal to this

one

so let's say um

this is your x2 y2

so this slope is equal to y minus

y naught divided by x minus x naught

you get this one and if you multiply

both sides by this expression

x minus x naught we get this

y minus y not equal to m times

x minus x naught equation

and we call that equation the

point-slope form the initial action a

point-slope form because meanwhile

point x not y-nut and the slope m

so take note of this this is very

important

and your next a very important number

discuss not then

slope intercept of form so melancholine

and intercept in particular y-intercept

number line

so let's say um this point is on the

line l so

homo observing the x coordinate is zero

meaning

b is the y in percent and using the

point slope form which is y minus y

naught equal to m

times x minus x naught

this is her x not y naught but bin laden

0 so x naught we will get this

expression or this equation and y

equal to f x plus b negative d

naught in a slope intercept form

this is a very important equation

from this equation

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the slope intercept form and the general

form so general form

so give it an opinion to find the

equation of

a line so let's say find the equation of

a line that passes through these

two points so he landed at a point and

we need

a slope and

and slope is y two minus y one

over x two minus x one e number one

point

slope you will get the equation of a

line so let's say this is

x one y one this is x to y two

so we get two minus

four divided by five minus

negative three so the slope is negative

two over

y naught equal to m times x minus x

naught

so i don't know equation

so for x not y naught it's just pick one

point

kites in a giant guy cp guide cq

uh pili in a language so this will serve

our x naught

and this is our y naught so x naught is

negative three

y naught is a four and the slope is

negative one fourth

so the equation is y minus

four equal to negative one fourth

x minus

negative 3

distributed

y equal to negative 1 for it

uh this is another way of doing it

direct and a y equal to m x plus b

slope negative one four so the equation

is y

equal to m x plus b or

y equal to negative one fourth x plus b

so allah means b to get the equation of

the length

so the doing one you just pick one point

either this one or this one

or for anonymous

so this particular point must satisfy

this equation because this point is on

the line

so at x y equal to five to one

a bus you get two equal to negative one

fourth x plus b

or two is negative five four x plus b

so b is two plus five fourths

or thirteen fourths so equation online

mo

is uh plug in attendee to c b

y equal to negative one fourth x plus 13

over four so we get the same

answer so you know two ways of doing

at least two ways of doing it just

in perpendicular lines so very important

to y equal to mx

plus because slope when you solve for y

the coefficient of x

is the slope of the line so we say that

two lines are parallel if

they have the same slope and if it's m

of

l one equal to m of l two the slope of

line one is this the slope of line two

is this

and two lines are perpendicular

if the products of their slope is

negative one

or whether it's

l1 is perpendicular to l2 okay or l1 and

l2 are perpendicular

l1 and l2 are parallel

so now of course an example

so find the slope intercept of form and

each one in

slope intercept form that is y equal to

m

x plus b of the equation of the line

that passes through one one and

perpendicular to l

so you know line nut and passes through

p

[Music]

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um y

y equal to two x plus

okay so you're missing line if i

yes using your slope num l

so the slope of l

observation y equal to m x plus b

so the slope of l is 2

and you want this missing line to be

perpendicular to l

so the slope of this missing line is

negative reciprocal of the slope of l

which is negative one half

so in equation and online is y equal to

negative one half x

plus b

equation this point

passes through this missing line you

know desired line at it

so meaning this point must satisfy this

equation

if you plug this particular x and y

in here the operating equation

so at x y

equal to 1 1 and the best we get one

equal to negative one half times one

plus

b or b is one plus one half

where b is three over so the equation of

the line

is y equal to negative one half x

theta and b is three over two

circles so let us define what a circle

is or

at least so a circle

is a set of points in the plane

equidistant from a fixed point

that fixed point is called the center so

in this case ethereum center net

blue red

highlights

[Music]

circle at imbabat point

little circle is equidistant to the

center

distance

[Music]

radius

so if the center is h k and the radius

is r

the equation is x minus h squared

plus y minus k squared equal to

r square so pana na ginga non explain

langnaden briefly

so let us recall the formula for the

distance between two points

so young distance between x1 y1

and x2 y2

is uh let's say d equal to

x2 minus x1 squared plus y2

minus y1 square okay

so let's say your circle now then

ayan and the center is h k

and the reduce is r so you want to

relate manga points

circle so let's say you have an

arbitrary point on the circle

x y so sabinate

circle and distance from the center

is r so we're going to use this distance

formula

so r is equal to the square root

of x minus

h squared

plus y minus k square

and voila when you square both sides of

this equation

when you expand this one tapas nila gay

musa left la ha

zero habilong said in general form

is something like x square plus y

squared plus

uh a constant times x

plus a constant times y plus a constant

equal to

zero so union general form or a general

equation and circle

all although in the obvious

center

regious form

because from the equation or from the

form itself you can find the results

in the center so from here

pano nyo malama new

or you can use the formula

uh h is equal to negative d

over 2 e is

negative sorry

k is

negative e over two so i need to divide

mulan atom by negative two to get

h at the demand divide by negative two

to get

k atom regions

so r square is equal to h square

plus k squared minus f so that is r

squared so go what i you know

examples so find

an equation for the circle with center

nato and passing through this point so i

know that your center religious form

so it's x minus h square plus y

minus k squared equal to r square

so our h k in this case is

two negative three so papa literally

nothing see h none two and k by negative

three

so x minus two squared plus

y minus negative magnitude plus

so say r squared

is a problem the circle passes through

negative one one so this negative one

one must satisfy this equation

so we'll plug in another nine

so we just replace x by negative one and

y by one

r square so r squared is equal to

negative three square kasama is negative

say any square so you get nine

using the man four square is sixteen

so r square is twenty five seventy one

so the equation of the circle is

equal to 25

so your new equation

so tangent lines to circles so

a line that intersects a circle at

exactly

one point is called a tangent line to

the circle

so example this one this line is tangent

to the circle because

it touches the circle at exactly one

point

and uh measuring special properties

the point of tangency is perpendicular

to your

tangent line okay

so as i illustrated in this figure

so perpendicular

perpendicular segment

so let us determine the slope intercept

form

of the equation of the line that is

tangent to the circle with this

equation at zero two

um provide time and sketch all the hindi

to accurate sketch huh

rough sketch lung so let's say that is

our

circle and this is

assambus is zero two let's say this is

zero two

circle so x square

plus y square

okay so h naught then

divide this by negative two you get one

our k is divide this by negative two you

get negative two

so r squared is h squared plus k

square minus f or

h squared is one plus four and f not

then is

negative twelve

so this is

so let's say

foreign

drawing

slope intercept form of the tangent line

so equation of tangent line

tl for tangent line is y equal to m x

plus b

[Music]

is

[Music]

is a line so b

is two so b is equal to b

so y equal to m x plus two

slope nalan ankula so this

gamma and tangent line so sabinate

if you connect the lines the

the points of tangency and the center of

the circle

so let's say it in blue line so

perpendicular

[Music]

for the slope of the blue line

okay so anion formula y minus y

2 minus negative 2 divided by

x minus x

or a negative four so

in the hand of nothing slope no tangent

line is a negative reciprocal of the

slope of the blue line

but the negative reciprocal of this is

one-fourth so you know nothing equation

of tangent line is

y equal to one-fourth x plus

plus two

and we can actually check this

and circle it type not then so it's x

squared

plus y squared minus

2x plus 4y

minus 12 equal to zero

so it must be x square plus y squared

up in a y equal to

[Music]

one-fourth x

one-fourth x plus two

yes as you can see ion sharp tangent

shall high

circle at point zero two so c a one zero

two

and that ends our discussions on lines

and circles so serena gets nino

goodbye and thank you