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Tangent to a Circle and Finding Line Intersections

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Ceri Thomas

29/04/2023

Maths

Coordinate geometry

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Coordinate geometry

Tangent to a Circle and Finding Line Intersections

A comprehensive guide to geometric concepts focusing on finding intersection of straight line graphs, equation of a tangent to a circle, and calculating perpendicular bisectors in geometry.

  • Explores fundamental concepts of straight line equations including gradients, y-intercepts, and parallel/perpendicular relationships
  • Details circle geometry including tangents, intersections, and center calculations
  • Covers advanced applications like finding circle equations from three points
  • Demonstrates practical problem-solving using simultaneous equations and geometric principles
  • Includes essential formulas for distance calculations and circle equations
...

29/04/2023

103

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Circle Properties and Equations

This section covers the fundamental properties of circles and their mathematical representations. The standard form equation (x-a)² + (y-b)² = r² is explored in detail.

Definition: A tangent is a line that touches the circle at exactly one point.

Vocabulary: The center coordinates are represented as (a,b), while r denotes the radius.

Example: When determining intersections between a line and circle, the discriminant (b²-4ac) indicates:

  • b²-4ac > 0: Two intersection points
  • b²-4ac = 0: One point (tangent)
  • b²-4ac < 0: No intersection
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Advanced Circle Applications

This section demonstrates practical applications of circle equations and intersection calculations. It includes detailed examples of finding intersection points and tangent equations.

Example: For the equation x² + (y-2)² = 29:

  1. Solve for intersection points using substitution
  2. Results in coordinates (2,7) and (-3,0)

Highlight: The perpendicular bisector of any chord always passes through the circle's center.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Circle Center Determination

This section explains methods for finding a circle's center using known points and perpendicular bisectors.

Definition: The perpendicular bisector method involves:

  1. Finding equations of perpendicular bisectors
  2. Calculating their intersection point
  3. This intersection determines the circle's center

Vocabulary: A diameter is any line segment passing through the center connecting two points on the circle.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Geometric Relationships and Applications

The final section synthesizes previous concepts to solve complex geometric problems involving circles and lines.

Highlight: The diameter (d) equals twice the radius (r), expressed as d = 2r.

Example: When finding a circle's equation from three points:

  1. Calculate perpendicular bisectors
  2. Find their intersection (center)
  3. Calculate radius using distance formula

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Subjects

Maths

Pure maths

Coordinate geometry

 

Maths

•

103

•

29 Apr 2023

•

5 pages

Tangent to a Circle and Finding Line Intersections

user profile picture

Ceri Thomas

@cerithomas

A comprehensive guide to geometric concepts focusing on finding intersection of straight line graphs, equation of a tangent to a circle, and calculating perpendicular bisectors in geometry.

  • Explores fundamental concepts of straight line equations including gradients, y-intercepts,... Show more
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

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Circle Properties and Equations

This section covers the fundamental properties of circles and their mathematical representations. The standard form equation (x-a)² + (y-b)² = r² is explored in detail.

Definition: A tangent is a line that touches the circle at exactly one point.

Vocabulary: The center coordinates are represented as (a,b), while r denotes the radius.

Example: When determining intersections between a line and circle, the discriminant (b²-4ac) indicates:

  • b²-4ac > 0: Two intersection points
  • b²-4ac = 0: One point (tangent)
  • b²-4ac < 0: No intersection
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the contentIt's free!

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By signing up you accept Terms of Service and Privacy Policy

Advanced Circle Applications

This section demonstrates practical applications of circle equations and intersection calculations. It includes detailed examples of finding intersection points and tangent equations.

Example: For the equation x² + (y-2)² = 29:

  1. Solve for intersection points using substitution
  2. Results in coordinates (2,7) and (-3,0)

Highlight: The perpendicular bisector of any chord always passes through the circle's center.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

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Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Circle Center Determination

This section explains methods for finding a circle's center using known points and perpendicular bisectors.

Definition: The perpendicular bisector method involves:

  1. Finding equations of perpendicular bisectors
  2. Calculating their intersection point
  3. This intersection determines the circle's center

Vocabulary: A diameter is any line segment passing through the center connecting two points on the circle.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Geometric Relationships and Applications

The final section synthesizes previous concepts to solve complex geometric problems involving circles and lines.

Highlight: The diameter (d) equals twice the radius (r), expressed as d = 2r.

Example: When finding a circle's equation from three points:

  1. Calculate perpendicular bisectors
  2. Find their intersection (center)
  3. Calculate radius using distance formula
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the contentIt's free!

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Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Straight Line Graphs Fundamentals

This section introduces the core concepts of straight line equations and their relationships. The fundamental equation y = mx + c forms the basis for understanding line gradients and intersections.

Definition: The gradient (m) represents the slope of a line, calculated using the formula m = (y₂-y₁)/(x₂-x₁).

Vocabulary: Y-intercept (c) is the point where a line crosses the y-axis.

Example: The length of a line segment can be calculated using the formula: length = √((y₂-y₁)² + (x₂-x₁)²)

Highlight: For perpendicular lines, the product of their gradients (m₁ × m₂) equals -1.

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Elisha

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This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user