EQUATION OF CIRCLE IN STANDARD FORM | PROF D

Summary

This educational video tutorial teaches viewers how to derive the standard form equation of a circle. It begins by explaining the concept of a circle and its basic elements, such as the center and radius. The presenter then illustrates the process using the distance formula, providing step-by-step examples with different centers and radii. The video concludes with a practical example involving a circle with a diameter defined by two points, showcasing the calculation of the center and radius before deriving the equation. The host, Prof D, encourages viewers to ask questions in the comments for further clarification.

Takeaways

• 📚 The video is an educational tutorial on finding the equation of a circle in standard form.
• 🌐 A circle is defined as the path of a point that moves at a constant distance from a fixed point, known as the center.
• 📏 The constant distance from any point on the circle to the center is called the radius.
• 📘 The standard form equation of a circle is derived from the distance formula and is given as \((x - h)^2 + (y - k)^2 = r^2\).
• 📍 The variables \(h\) and \(k\) represent the coordinates of the circle's center, and \(r\) is the radius.
• 🔍 Example 1 demonstrates finding the equation of a circle with a center at (0,0) and a radius of 5, resulting in the equation \(x^2 + y^2 = 25\).
• 📐 Example 2 shows the process for a circle with a center at (-4,5) and a radius of 4, leading to the equation \((x + 4)^2 + (y - 5)^2 = 16\).
• 📈 Example 3 involves calculating the equation of a circle given the endpoints of its diameter, resulting in the equation \((x - 1)^2 + (y - 4)^2 = 10\).
• 🧭 The midpoint formula is used to find the center of the circle when the diameter's endpoints are known.
• 📝 The video emphasizes the importance of correctly identifying the center and radius to derive the circle's equation.
• 👋 The presenter, Prof D, encourages viewers to ask questions or seek clarifications in the comments section.

Q & A

What is the definition of a circle according to the video?

A circle is the path or locus of a point that moves at a constant distance from a fixed point, called the center.

What is the constant distance from any point on the circle to the center called?

The constant distance of any point from the center is called the radius.

What is the standard form of the equation of a circle?

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

In the video, what is the first example of finding the equation of a circle?

The first example is finding the equation of a circle with a center at (0, 0) and a radius of 5.

What is the equation of the circle in the first example after substituting the values for h, k, and r?

The equation is x² + y² = 25 after substituting h = 0, k = 0, and r = 5.

What is the second example's circle center and radius according to the video?

The second example has a circle with a center at (-4, 5) and a radius of 4.

How is the equation of the circle in the second example simplified?

The equation simplifies to (x + 4)² + (y - 5)² = 16 after substituting h = -4, k = 5, and r = 4.

In the third example, how are the end points of the diameter given?

The end points of the diameter in the third example are given as (4, 5) and (-2, 3).

What is the method used in the third example to find the center of the circle?

The midpoint formula is used to find the center of the circle in the third example.

What is the final equation of the circle in the third example after simplification?

The final equation is (x - 1)² + (y - 4)² = 10 after applying the midpoint formula and the distance formula to find h, k, and r.

What does the video suggest for viewers who have questions or need clarifications?

The video suggests that viewers should put their questions or clarifications in the comment section below.

Outlines

00:00

📚 Introduction to Circle Equations

This paragraph introduces the topic of the video, which is finding the equation of a circle in standard form. The presenter explains the concept of a circle as the path of a point that maintains a constant distance from a fixed point, known as the center. The constant distance is referred to as the radius. The standard form of a circle's equation is given as (x - h)² + (y - k)² = r², where (h, k) are the coordinates of the center and r is the radius. An example is then introduced to illustrate how to derive this equation.

05:00

🔍 Example 1: Circle with Center at (0,0) and Radius 5

The first example demonstrates how to find the equation of a circle with a center at the origin (0,0) and a radius of 5. The presenter identifies the center coordinates (h, k) and the radius r, then substitutes these values into the standard equation of a circle. Simplifying the equation leads to x² + y² = 25, which represents a circle with a radius of 5 units extending in all directions from the origin.

10:02

📐 Example 2: Circle with Center at (-4,5) and Radius 4

In the second example, the presenter shows how to calculate the equation of a circle with a center at (-4,5) and a radius of 4. The process involves identifying the center coordinates and the radius, then substituting these into the standard circle equation. After simplification, the equation becomes (x + 4)² + (y - 5)² = 16, indicating a circle 4 units in radius centered at the given point.

15:04

📏 Example 3: Circle from Diameter Endpoints

The third example involves finding the equation of a circle when given the endpoints of its diameter. The presenter uses the midpoint formula to determine the center of the circle and the distance formula to calculate the radius. The final equation, derived from the calculated center (1,4) and radius, is (x - 1)² + (y - 4)² = 10, representing a circle with a radius of the square root of 10 units.

👋 Conclusion and Sign Off

The video concludes with the presenter summarizing the content and inviting viewers to ask questions or seek clarifications in the comments section. The presenter, identified as Prof D, thanks the viewers for watching and signs off with a friendly farewell, indicating the end of the educational content.

Keywords

💡Circle

A circle is defined as the set of all points in a plane that are at a constant distance from a fixed point, known as the center. In the video, the concept of a circle is central to explaining how to derive the equation of a circle. The script uses the circle's definition to introduce the standard form equation of a circle, emphasizing the constant distance, or radius, from the center to any point on the circle.

💡Locus

Locus refers to the path described by a moving point that satisfies a particular condition. In the context of the video, the term is used to describe the path of a point that moves at a constant distance from the center, thus forming a circle. This concept is foundational in understanding the geometric nature of a circle.

💡Center

The center of a circle is the fixed point from which all points on the circle are at an equal distance, known as the radius. The script mentions the center when explaining the standard form of a circle's equation, using it as a reference point (h, k) for the equation (x - h)^2 + (y - k)^2 = r^2.

💡Radius

The radius is the constant distance from the center of a circle to any point on the circumference. It is a key component in the script's explanation of the circle's equation, where 'r' represents the radius and is squared to form part of the equation (x - h)^2 + (y - k)^2 = r^2.

💡Standard Form

The standard form of a circle's equation is a specific way to express the relationship between the circle's center and its points. In the video, the standard form is introduced as (x - h)^2 + (y - k)^2 = r^2, which is used to find the equation of a circle given its center and radius.

💡Distance Formula

The distance formula is used to calculate the distance between two points in a plane. In the script, it is mentioned as the basis for deriving the standard equation of a circle, where 'd' represents the distance, calculated as sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2).

💡Example

Examples are used throughout the script to illustrate how to apply the concepts and formulas discussed. The video provides three examples, each demonstrating the process of finding the equation of a circle with different centers and radii, helping viewers understand the practical application of the theory.

💡Midpoint Formula

The midpoint formula is used to find the midpoint of a line segment, which in the context of the video, helps in determining the center of a circle when the endpoints of a diameter are known. The formula is applied in example three to calculate the center (h, k) as the average of the x-coordinates and y-coordinates of the endpoints.

💡Diameter

The diameter of a circle is any line segment that passes through the center and connects two points on the circle. In the script, the diameter is used in example three to find the center of the circle by using the midpoint of its endpoints.

💡Equation Derivation

Equation derivation in the video refers to the process of obtaining the standard form equation of a circle from given conditions. The script demonstrates this process through examples, showing how to identify the center and radius and then apply them to the formula (x - h)^2 + (y - k)^2 = r^2.

💡Prof D

Prof D appears to be the presenter or the host of the video, guiding viewers through the mathematical concepts and examples. The name is used to establish a personal connection with the audience and to sign off at the end of the video.

Highlights

Introduction to the concept of a circle as the path of a point moving at a constant distance from a fixed point, the center.

Explanation of the constant distance from any point on the circle to the center, known as the radius.

Derivation of the standard form equation of a circle given the center coordinates (h, k) and radius r.

Use of the distance formula as the basis for the standard equation of a circle.

Example 1: Finding the equation of a circle with a center at (0,0) and a radius of 5.

Substitution of values into the standard equation to find the equation of a circle with a specific center and radius.

Simplification of the equation to show the circle's equation as x^2 + y^2 = 25.

Geometric interpretation of the circle's equation, indicating a 5-unit radius in all directions from the center.

Example 2: Determining the equation of a circle with a center at (-4,5) and a radius of 4.

Identification of the center coordinates and radius for the second example circle.

Application of the standard equation formula with the given values for h, k, and r.

Simplification resulting in the circle's equation as (x + 4)^2 + (y - 5)^2 = 16.

Example 3: Using the diameter's endpoints to find the equation of a circle.

Utilization of the midpoint formula to find the center of the circle when given the diameter's endpoints.

Application of the distance formula to calculate the radius of the circle from its diameter.

Final equation derivation for the circle with endpoints (4,5) and (-2,3) resulting in (x - 1)^2 + (y - 4)^2 = 10.

Conclusion of the video with an invitation for questions and further discussion in the comments.

Transcripts

[Music]

hello guys welcome back to my channel

in this video i will show you how to

find the equation of a circle

in standard form so background at the

final

anatomy what is a circle

so a circle is the path or locus of a

point

that moves at a constant distance from a

fixed point

called the center so they are adding

center

okay the constant distance of any point

from the center

is called the reduce

so the equation of a circle in standard

form

so given a circle with radius r

and with the center at c h

comma k and let p

of x y be any point on the circle

then by definition we have x minus

h square plus y minus k

squared is equal to r squared

is the equation of a circle with center

at c

h k and reduce r

so your equation nothing or your

standard

equation and circle is not derived from

the distance formula

where d is equal to the square root of

x sub 2 minus x sub 1

square plus y sub two

minus y sub one

square okay where you adding h and k

is young x sub one and y

sub one so let's have example number one

find an equation of the circle with the

following conditions

so number one center is at zero

zero and reduce is equal to

five so first identify muy not nch

and k so alumni

h and k is um coordinates now center

nothing which is h naught and

is young x ordinate and that is zero

and young k naman nathan is your y

ordinate which is

zero and your age is nothing is equal to

five so after nothing my identifies the

h k and r so i did not think that

meeting you

uh four million atmos attacks which is

x minus h squared

plus y minus k

squared is equal to r

squared so on on the other guys

is is the subject substitute log not a

new value in

h and k which is both zero

so that is x minus zero

square plus y

minus 0 squared equals

r which is 5

squared all right then simplify nothing

so that

is we have x minus 0 that is x then

square plus y minus zero

which is y then square

equals five square and that is

twenty-five so ethanol

which is five units so on the govinda

nandito guys says magbibilan

on five units going to the right and to

the left

then five units

so start is we have one two

three four five so nandita atting

first point then five tai pata

so we have five

then another five papunda is a left side

so this is

negative five then five seven but

you have negative five okay then after

nothing my platform

uh the target nothing points is catching

you along guys

young adding circle

so next let's have example number two

center is at negative four comma five

and reduce is equal to four

so again on first step nathan is

identifying when athens say

h k and

r so in this case

center and

h corresponds to negative four

and your k not n is equal to five

then of course our radius is equal to

positive four after nothing identifying

a given fed in the nothing gammite

uh formula standard equation a circle

which is

x minus h squared

plus y minus

k squared equals

r squared

next is a substitute

value so h is negative four

so that is x minus negative four

then square plus y

minus k and k is equal to 5

squared equals r

squared and that is 4

squared okay

next is simplifying nothing so we have

two negatives

so negative times negative that is

positive four then square

plus we have y minus five so

nothing young

then four squared is equal to

sixteen so

you adding negative four then i'm i

think k

is five so one two

three four five ninety two my onion

center okay i'm sorry so this is

positive five

so upward dial okay so one two three

four

five bit on ion center

okay so next is

after nothing young center so again

four units to the right to the left

and going up and down

so one two three four three i'm not in

first point

one two three four then one

two three four then

one two three four seven

so next let's have example number three

so diameter with end points at four

five and negative two

three so example

given since hindi given young center

plot nothing in four or five so that is

nandito

okay so you know what in four five

next is you negative two three so that

is here so this is negative two

three okay so this will be now

the diameter no adding circle

okay so alumni

midpoint formula okay

so to solve for h

and since h is x ordinate num center

so that is four plus negative two

divided by two all right so to your

midpoint formula

so that is four plus negative two

that is two then divided by two we have

one

okay then for kinaman

since yonke is in y ordinate

and gaga meeting is y sub one and

y sub two which is 5 3

so 5 plus 3 that's 8

divided by 2 we have k

equals 4 so on location

outing center is one

four so nothing

okay

center

okay so applying the distance formula

so that is r equals square root of

so gaga meeting nothing you distance

four million at n

x sub two minus x sub one

square plus y sub 2

minus y sub 1 squared

so in our case if you're adding x sub 1

y sub 1

then x sub 2 y sub 2.

okay so substitute not the number given

so that is uh we have x sub 2 that is 4

minus 1 square plus

y sub 2 which is 5 minus y sub 1 which

is 4 then square

so we have 4 minus 1 squared that is 3

then squared that is 9. plus

five minus four is one then square we

have

one so language is not n is equal

to square root of ten

standard form now

circle which is x minus h

squared plus y

minus k square equals r

squared so substitute nothing you having

h

k and r so that is

x minus 1 squared

plus y minus k which is 4

squared equals square root of 10

then squared so simplify along nothing

square root of 10 squared so in this

case

square so final answer nothing we have

x minus one square

plus y minus four squared

equals 10.

equation and circle in standard form

okay this is the end of our video

i hope mina tunan chao class so if you

have questions or clarifications kindly

put them in the comment section below

so thank you guys for watching this is

prof d

i'll catch you on the flip side bye

[Music]