Learning Materials
Features
Discover
Coordinate geometry describes everything related to the Cartesian plane and is therefore sometimes known as Cartesian geometry. Remember x and y coordinates? The Cartesian plane is the two-dimensional plane formed by the intersection of x and y.
Millions of flashcards designed to help you ace your studies
Sign up for freeWhat is another name for coordinate geometry?
What are the two axes that form the Cartesian plane?
What is the key feature of straight line graphs in coordinate geometry?
Which section of coordinate geometry involves finding the radius and center of a circle?
What method can help find a tangent to a circle?
What variable is often used in parametric equations?
What do parallel lines have in common?
What is the product of gradients of perpendicular lines?
What is the equation form for a circle with radius r and center (a, b)?
What is required to model the amount of money an ice-cream van makes as a straight line graph?
What is another name for coordinate geometry?
What are the two axes that form the Cartesian plane?
What is the key feature of straight line graphs in coordinate geometry?
Which section of coordinate geometry involves finding the radius and center of a circle?
What method can help find a tangent to a circle?
What variable is often used in parametric equations?
What do parallel lines have in common?
What is the product of gradients of perpendicular lines?
What is the equation form for a circle with radius r and center (a, b)?
What is required to model the amount of money an ice-cream van makes as a straight line graph?
Achieve better grades quicker with Premium
Geld-zurück-Garantie, wenn du durch die Prüfung fällst
Did you know that StudySmarter supports you beyond learning?
Review generated flashcards
Start learning or create your own AI flashcards
Team Coordinate Geometry Teachers
Coordinate geometry is a very important study as it allows us to develop graphical representations for different things such as parallel and perpendicular lines and curves we couldn't normally graph.
We split Coordinate Geometry into three key sections:
Let's look at these in a bit more detail.
In order to understand coordinate geometry, we will look at straight line graphs in a lot of detail, starting with calculating gradients and intercepts. Then we will move on to parallel and perpendicular lines. Finally, we will start modeling using straight line graphs.
Here's is an example of a question involving straight line graphs. This question will require calculating the gradient.
The amount of money an ice-cream van makes in a day can be modeled as \(y = 5s-12\). Where s is the amount of ice creams sold and y is the amount of money made in pounds.
Find the price of each ice cream.
Calculate the amount of ice cream that needs to be sold so that the ice cream van doesn't make a loss.
SOLUTION: 1. The gradient of this line is the money made from sales. Remember, in a graph, m is the gradient.
\(y = mx+c\)Therefore the gradient of this graph is 5. So each sale is £5. To not make a loss \(5s-12 \geq 0\) We can solve this by saying \(5s \geq 12\)
Circles are an important part of coordinate geometry. We can use information about circles along with other theories of coordinate geometry to solve more complicated problems.
Remember, a circle with radius r and center (a, b) has an equation: \((x-a)^2 + (y-b)^2 = r^2\)
A circle has an equation \((x-2)^2 + (y-4)^2 = 25 \qquad (5,8)\)
SOLUTION: First, we need to find the equation of the radius. The centre of the circle is (2, 4), the point we are concerned about is. Find the gradient:\(\frac{8-4}{5-2} = \frac{4}{3}\)So general equation of radius is \(y = \frac{4}{3}x +c\) . So where c is some constant is the equation of the radius. Now we use the fact that the tangent is perpendicular to the radius. When two lines are perpendicular to their gradients product -1 . So the gradient of the radius and the gradient of the tangent must multiply to make -1 . Therefore if we call the gradient of the tangent p\(\frac{4}{3}p = -1p = -\frac{3}{4}\)So our equation is:\(y = -\frac{3}{4}x+c\)To find our constant c simply substitute values \(x =5 \text{ and }y = 8\)\(8 = -\frac{3}{4}(5)+c8 = -\frac{15}{4} + cc = 11.75\)So our final answer is:\(y = -\frac{3}{4}x + 11.75\)This is a graphical representation of the circle and perpendicular line:
Parametric equations represent everything in terms of one variable. The variable normally used is t.
This is because there are a lot of more complicated equations where it is better to represent each x and y in terms of the same variable.
Here's an example of a set of parametric equations.
\(x = 2\cos(t); \space y = 2\sin(t)\)This is the parameterization of a circle as:
\(x^2+y^2 = (2\cos(t))^2 + (2\sin(t))^2 = 4 \cos^2(t) + 4 \sin^2(t) = 4(\sin^2(t) + \cos^2(t)) = 4(1)= 4\)
Below is an example of a parametric equations question.
A curve C contains the following parametric equations.
\(x = 4\cos(t+\frac{\pi}{6}); \space y = 2\sin(t)\)Prove that \(x + y = 2\sqrt3 \cos(t)\)
Show that the Cartesian equation of C is \((x+y)^2 +ay^2 = b\) where a and b are constants to be found.
SOLUTION:
Well \(x+y = 4\cos(t+\frac{\pi}{6}) + 2\sin t\).
By addition formula
\(4 \cos(t+\frac{\pi}{6}) = 4\cos(t)\cos(\frac{\pi}{6})-4\sin(t)\sin(\frac{\pi}{6})4\cos(t + \frac{\pi}{6}) = 4\frac{\sqrt3}{2} \cos(t) 2\sin(t) 4\cos(t+\frac{\pi}{6}) = 2\sqrt{3} \cos(t) -2\sin(t)4\cos(t+\frac{\pi}{6}) +2\sin(t) = 2\sqrt3 \cos(t) - 2\sin(t)+2\sin(t) = 2 \sqrt3 \cos(t)\)
2.
\((x+y)^2 = (2\sqrt3 \cos(t))^2 = 12 \cos^2{t}y^2 = 4\sin^2{t} 12\cos^2{t}+4a(\sin^2{t}) = b\)
By \(\sin^2{t} + \cos^2{t} = 1\):
\(12\cos^2{t}+12\sin^2{t} = 124a = 12 \rightarrow a = 3b =12\)
Straight line graphs are decided by a gradient and the y-intercept.
Parallel and perpendicular lines are decided by gradients.
Parallel lines contain the same gradient.
Perpendicular lines have gradients which product -1.
Circle theorems can be used to help find equations of lines on a Cartesian plane.
Coordinate geometry ties together geometrical concepts and rules of lines in Cartesian coordinates.
Parametric Equations involve writing everything in terms of one variable.
Sign up for free to gain access to all our flashcards.
What is coordinate geometry?
Coordinate geometry is the study of the Cartesian plane.
What is a straight line graph?
A graph with the equation y=mx+c.
What is a set of parallel lines?
Two lines that contain the same gradient and never meet.
What is a tangent perpendicular to in a circle?
A radius.
At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.
Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.
Get to know Lily
Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.
Get to know Gabriel
Vaia is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.
Learn moreSave explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of Vaia.
Already have an account? Log in
Sign up to highlight and take notes. It’s 100% free.
Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App 
