# Coordinate geometry in the (x,y) plane

**AS Level**

- Understand and use the equation of a straight line, including the forms y -y
_{1}= m(x-x_{1}) and ax+by+c=0 and ; gradient conditions for two straight lines to be parallel or perpendicular - Be able to use straight line models in a variety of contexts
- Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x-a)
^{2}+ (y-b)^{2}=r^{2}; completing the square to find the centre and radius of a circle; use of the following properties:

• the angle in a semicircle is a right angle

• the perpendicular from the centre to a chord bisects the chord

• the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point

**A Level**

- Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms
- Use parametric equations in modelling in a variety of contexts

### Coordinate Geometry

This resource is seven Rich Starting Point activities covering a range of topics, each one having some activity which explores coordinate geometry. They are accompanied by teachers' notes. These two are concerned with circles.

Circle Property: Students generate two coordinates. The coordinates form the end points of the diameter of a circle. Students have to find the equation of the circle formed, compare their results with colleagues and explain their findings.

Circles or Not?: Students use a graph plotter to alter the coefficients of the equation of a circle and explore which values produce a circle and why.

### Geometry

The first of these files provides revision of straight line graphs; the equation, gradient, intercepts .

It contains brief notes, examples with solutions and also brief exercises with answers.

### Geometry

The topics covered include the geometry of a circle, the gradient and intercept of straight lines and properties of straight line segments.

Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples.

### Parametric Equations

For success with this resource students would need to understand differentiation of a quadratic function.

This RISP starter activity is in the form of a puzzle in which students are given a general parametric equation with missing coefficients. A number of clues are given such as a point through which the curve passes. Students have to use their understanding of parametric equations to find the missing numbers.

### Circles

This resource contains eight problems requiring students to explore the equations of a variety of circles. Many of the problems are open-ended and provide the opportunity for discussion as students attempt to justify their solution.

Students are required to explain whether points lie inside or outside a given circle and whether two points are the two ends of the diameter to a given circle. Students are required to complete the square in order to manipulate the equation of a circle.

### Circle Geometry

This resource is a set of 3 lesson ideas which explore the properties of circles and their equations.

They deal with finding the radius and coordinates of the centre of circles and include completing the square.

### Connecting Perpendicular Lines A10

In this resource from the DfE Standards Unit students explore straight lines. They identify perpendicular gradients and lines that are perpendicular and explain the reasons why lines are parallel and perpendicular.

### Straight Lines

This resource consists of 2 interactive Excel files which allow students to explore parallel and perpendicular lines. Students can explore gradient and intercepts of straight lines and the relationship between the gradients of perpendicular lines.

### Straight Line Graphs

This resource contains seven problems requiring students to think, explore, explain and justify the mathematics they use in order to solve the problems presented.

Students will explore the equations of straight lines, parallel lines, perpendicular lines, equations of straight lines in a variety of forms and to find possible equations of lines which do not pass through a specified point.

### Straight Lines: Y = mX + c

This interactive Excel program concerns the equation of straight lines in the form ax+by+c=0. The first two sheets deals with the equation of the line through two given points and the equation of the line through a point perpendicular to a given line. The program allows the graphs of the lines to be shown on coordinate grids.

Further sheets deal with similar problems including finding the equation of the perpendicular bisector of a line segment, finding the coordinates of the circum-centre of a triangle and the equations of lines relating to a circle.

The resource also has a printable set of questions about finding the equation of the line through two given points.

### Linear Functions

This resource consists of three files which contain a number of activities designed to provide practise and to strengthen understanding of linear functions.

"The properties of linear functions" deals with the gradients and intercepts of lines,

"Parallel and perpendicular" explores the properties of parallel and perpendicular lines.

The third file deals with finding the perpendicular bisector given the coordinates of two points.