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Coordinate Geometry

Coordinate Geometry

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.

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:

STRAIGHT LINE GRAPHS - Understanding how gradients work and how we can use this in modeling. As well as understanding gradients of parallel and perpendicular lines.

CIRCLES - Understand how algebraic methods such as completing the square can help us find the radius and center of a circle. Also understanding how to find a tangent to a circle using methods adopted from straight line graphs.

PARAMETRIC EQUATIONS - Understanding how we can use one variable to describe what two variables do and understanding how we can find equations for graphs we normally wouldn't be able to find by just looking at the graph.

Let's look at these in a bit more detail.

Straight line graphs

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

. Where is the amount of ice creams sold and is the amount of money made in pounds.

  1. Find the price of each ice cream.

  2. 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.

    Coordinate Geometry, Straight Line Graph, StudySmarter

Therefore the gradient of this graph is 5. So each sale is £ 5. To not make a loss We can solve this by saying Therefore: So at least 3 sales must be made.

Circles

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:

Here is an example of a circle question at A-Level.

A circle has an equation . Find the equation of the tangent at

SOLUTION: First, we need to find the equation of the radius. The center of the circle is , the point we are concerned about is. Find the gradient:So general equation of radius is . 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 . So the gradient of the radius and the gradient of the tangent must multiply to make . Therefore if we call the gradient of the tangent So our equation is:To find our constant simply substitute valuesSo our final answer is:

This is a graphical representation of the circle and perpendicular line:Coordinate geometry, Graphical representation of a circle and perpendicular line, StudySmarter

A graph of a circle and the tangent line

Parametric equations

Parametric equations represent everything in terms of one variable. The variable normally used is .

This is because there are a lot of more complicated equations where it is better to represent each and in terms of the same variable.

Here's an example of a set of parametric equations.

This is the parameterization of a circle as:

Below is an example of a parametric equations question.

A curve C contains the following parametric equations.

  1. Prove that

  2. Show that the Cartesian equation of C is where a and b are constants to be found.

SOLUTION:

  1. Well .

By addition formula

2.

By

Coordinate Geometry - Key takeaways

    • 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.

Frequently Asked Questions about Coordinate Geometry

Coordinate geometry is the study of the Cartesian plane.

A graph with the equation y=mx+c.

Two lines that contain the same gradient and never meet.

A radius.

Final Coordinate Geometry Quiz

Question

What is the diameter of the circle?

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Answer

The distance from one side of the circle to another going through the centre.

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What is the circumference of a circle?

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Answer

The total perimeter of a circle.

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What is the radius of a circle?


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The distance between the centre of the circle and the circumference.

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What is a sector?

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An area bordered by two radii.

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What is a chord?


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A line that goes from one side of circumference to the other without going through the centre.

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What is a segment?


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An area bordered by a chord or a circumference.

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What is a tangent?


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Answer

A line outside of the circle that touches the circumference of a circle at one point.

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 What is an arc?


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A section of the circumference.

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What does it mean if something is perpendicular?

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Answer

That it intersects with something at a right angle

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What does it mean when something bisects?

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It intersects a line through the middle, and the line on either side is the same length.

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What are the two things you need to substitute into the linear formula to create an equation for the perpendicular bisector?


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Answer

The point of bisection (midpoint) and the gradient of the perpendicular bisector.

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How do you calculate the perpendicular gradient?

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Answer

You need to do the inverse reciprocal of the original gradient.

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Question

A line segment from (7, 7) to (22, -14) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

 y = 0.71x - 13.795

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Question

A line segment from (-10, -10) to (-1, -1) is perpendicular bisected by line 1. What is the equation for line 1?


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Answer

y = -1x - 11

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A line segment from (2, -7) to (12, -19) is perpendicularly bisected by line 1. What is the equation for line 1?

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Answer

y = 0.83x - 18.81

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A line segment from (-8, -6) to (2, 4) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

y = -1x - 4

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A line segment from (0, 0) to (5, -20) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

0.25x - 10.625

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A line segment from (-4, -3) to (-1, 3) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

 y = -0.5x - 1.25

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A line segment from (2, 7) to (14, -1) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

y = 1.5x - 9

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A line segment from (-10, 10) to (5, -1) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

y = 1.36x + 7.9

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A line segment from (-2, 7) to (8, -18) is perpendicularly bisected by line 1. What is the equation for line 1?


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y = 0.4x - 6.7

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A line segment from (-10, 7) to (-1, -1) is perpendicularly bisected by line 1. What is the equation for line 1?


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y = -1.83x + 12.235

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A line segment from (-11, 5) to (-2, 20) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

y = -0.6x + 8.6

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A line segment from (2, 5) to (6, 15) is perpendicularly bisected by line 1. What is the equation for line 1?


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Answer

y=0.4x+11.6

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A line segment from (5,9) to (12, -5) is perpendicularly bisected by line 1. What is the equation for line 1?

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Answer

y=0.5x-2.25

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What is a sector?

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Answer

A sector of a circle is a proportion of a circle where two slides are radii

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Question

How do you find the sector of a circle?

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Answer

To find the sector of a circle you need to use one of the formulas for the area of the sector. Which one you use is dependent on whether the angle at the centre is in radians or in degrees

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There are two formulas to find the area of a sector of a circle. What decides which one you use?

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Answer

Whether the angle at the centre of the circle is in degrees or radians

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What is an arc length?


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Answer

A proportion of the circle’s circumference, which is defined by the sector.

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Circle A has a diameter of 10cm. A sector of circle A has an angle of 120. What is the area of this sector? 


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Answer

105 (3 s.f)

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Circle B has a diameter of 4cm. A sector of circle B has an angle of 20. What is the area of this sector? 


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Answer

2.79 (3 s.f)

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Circle D has a radius of 10cm. A sector of circle D has an angle of 60. What is the arc length of this sector? 


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Answer

10.5 (3 s.f.).

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Circle E has a diameter of 15cm. A sector of circle E has an angle of 90. What is the arc length of this sector? 


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Answer

11.8 (3 s.f.)

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What are radians?


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Answer

An alternative to degrees for measuring angles.

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Question

Circle F has a diameter of 9cm. A sector of circle E has an angle of 115. What is the arc length of this sector?


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Answer

9.03 (3 s.f.)

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Question

Circle G has a radius of 5cm. Within Circle G, there is a sector with an angle of 1.2 radians. What is the area of this sector?


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Answer

15

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Circle H has a radius of 11cm. Within Circle H, there is a sector with the angle x radians and an area of 222.9cm2. What is the value for the angle of this sector?


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Answer

3.8 radians

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Question

What is a tangent?

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Answer

A tangent is a line which touches the circle at a specific point

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What are the steps to finding the tangent of a circle at a specific point?

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Answer

Find the gradient of the radius using the centre of the circle and the point of interception; find the gradient of the tangent using the radius; using a linear equation formula, the gradient of the tangent and the point of interception, create a linear equation for the tangent 

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Question

What does the radius of a circle act as when we are finding the tangent?


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Answer

A normal line perpendicular to the tangent

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Question

How do you find the gradient of the tangent using the gradient of the radius of the circle?


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Answer

Find the negative reciprocal of the gradient of radius of the circle

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Question

Circle B has the equation (x+4)2+(y+2)2=10. A tangent touches Circle B at point (-1, -1). What is the equation of the tangent?


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Answer

 y=3x+2

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Circle C has the equation (x-3)2+(y-2)2=20.  A tangent touches Circle C at point (7, 4). What is the equation of the tangent?


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Answer

y=-2x+18

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Circle D has the equation (x-3)2+(y-2)2=20.  A tangent touches Circle D at point (7, 0). What is the equation of the tangent?


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Answer

y=2x-14

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Question

Circle E has the equation (x+1)2+(y+1)2=18. A tangent touches Circle E at point (2, 2). What is the equation of the tangent?


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Answer

y=-x+4

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Question

Circle F has the equation (x+1)2+(y+1)2=18. A tangent touches Circle F at point (2, -4). What is the equation of the tangent?


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Answer

y=x-6

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Question

Circle G has the equation (x-5)2+(y+2)2=8. A tangent touches Circle G at point (7, 0). What is the equation of the tangent?

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Answer

y=x+7

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Question

Circle H has the equation (x-5)2+(y+2)2=8. A tangent touches Circle H at point (7, -4). What is the equation of the tangent?


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Answer

 y=x-11

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Question

Circle I has the equation (x-4)2+(y-6)2=5. A tangent touches Circle I at point (2, 5). What is the equation of the tangent?


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Answer

y=-2x+9

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Circle K has the equation x2+y2=32. A tangent touches Circle J at point (4, -4). What is the equation of the tangent?


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Answer

y=x-8

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