EDEXCEL A-LEVEL PURE MATHS (6): CIRCLES

How can you find the midpoint of a line segment?

- You can find the midpoint of a line segment by averaging the x and y-coodinates of its endpoints.

- The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is:

((x₁ + x₂)/2, (y₁ +y₂)/2).

What is an example of finding the midpoint of a line segment?

What is an example of finding coordinates using the equation ((x₁ + x₂)/2, (y₁ +y₂)/2)?

What is the perpendicular bisector of a line segment?

- The perpendicular bisector of a line segment, AB, is the straight line that is perpendicular to AB and passes through the midpoint of AB.

- If the gradient of AB is m, then the gradient of the perpendicular bisector will be -1/m.

What is an example of finding the equation of the perpendicular bisector to a line, given two points the line passes through?

What is a circle and what is the equation of a circle?

- A circle is the set of points that equidistant from a fixed point.

- You can use Pythagoras' theorem to derive the equation of circles on a coordinate grid.

- For any point (x, y) on the circumference of a circle, you can use Pythagoras' theorem to show the relationship between x, y and the radius r.

- The equation of a circle with centre (0, 0) and radius, r, is x² + y² = r².

- When a circle has a centre (a, b) and radius r, you can use the following general form of the equation of a circle: (x - a)² + (y - b)² = r².

What is an example of finding the equation of a circle when given a centre (a, b) and a radius, r?

What is an example of finding the equation of a circle and a point (a, b) it passes through when given a an equation in the form (x - a)² + (y - b)² = r²?

What are more complex problems involving circle equations?

What is the other form the equation of a circle can be given in?

- You can multiply out the brackets in the equation of a circle to find it's alternate form:

1) (x - a)² + (y - b)² = r²

2) x² - 2ax + a² + y² - 2by + b² = r²

3) x² + y² - 2ax - 2by + b² + a² - r² = 0

- The equation of a circle can be given in the form: x² + y² + 2fx + 2gy + c = 0.

- This circle has centre (-f, -g).

- This circle has radius √f² + g² - c.

- If you need to find the centre and radius of a circle with an equation in expanded form, it's usually safest to complete the square for the x and y terms.

What is an example of completing the square of the x and y terms to find the centre and radius of a circle with an equation in expanded form?

How do you find the coordinates of intersections of a straight line and a circle?

- By using algebra.

- A straight line can intersect a circle once, by just touching a circle, or twice.

- Not all straight lines will intersect a given circle.

What is an example of finding the coordinates of intersections of a straight line and a circle?

What is an example of proving there are no intersections of a straight line and a circle?

What are tangents and chords and what are their properties?

- A tangent to a circle is a straight line that intersects the circle at only one point.

- A tangent to a circle is perpendicular to the radius of the circle at the point of intersection.

- A chord is a line segment that joins two points on the circumference of a circle.

- The perpendicular bisector of a chord passes through the centre of a circle.

- You can use the properties of tangents and chords within circles to solve geometric problems.

What is an example of using the properties of tangents within circles to solve geometric problems?

What are triangles and how do they form circumcircles?

- A triangle consists of three points called verticies.

- It's always possible to draw a unique circle through the three verticies of any triangle.

- This circle is called the circium circle of the triangle.

- The centre of the circle is called the circumcentre of the triangle and is the point where the perpendicular bisectors of each side intersect.

What are the rules with triangles inside circles?

- For a right-angled triangle, the hypotenuse of the triangle is a diamter of the circumcircle.

- Therefore, if angle PRQ = 90°, then R lies on the circle with diamter PQ.

- Therefore, the angle in a semicircle is always a right angle.

How do you find the centre of a circle given any three points on the circumference?

- Find the equations of the perpendicular bisectors of two different chords.

- Find the coordinates of the point of intersection of the perpendicular bisectors.

What is an example of a question involving proof that a straight line is a diameter of a circle?

EDEXCEL A-LEVEL PURE MATHS CHAPTER SIX: CIRCLES

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EDEXCEL A-LEVEL PURE MATHS CHAPTER SIX: CIRCLES

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EDEXCEL A-LEVEL PURE MATHS CHAPTER SIX: CIRCLES

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