Circles - [Pure 3].

Question 1

How can you best describe a circle?

Answer 1

A series of points which are equidistant from a fixed centre.

Question 2

What is the General Form Equation of a Circle?

Answer 2

(x - a)² + (y - b)² = r².

Question 3

What is the Transformation from x² + y² = r², to (x - a)² + (y - b)² = r²?

Answer 3

A Translation by vector [a b].

Question 4

How do you find the centre and radius of a circle from an equation in the form: x² + y² + 2ax + 2by + c = 0?

Answer 4

Complete the Square.
* ((x + a)² - (2a)²) + ((y + b)² - (2b)²) + c = 0.

goes to…

• (x + a)² + (y + b)² = d.

Question 5

When a line and a circle meet how many different roots can there be?

Answer 5

0 (no roots - don’t intersect), 1 (repeated root) or 2 (different roots).

Question 6

What can you use to work out how many times a line and a circle intersect?

Answer 6

b² - 4ac.

Question 7

For ax² + bc +c = 0, what do you find for no roots?

Answer 7

b² - 4ac < 0.

Question 8

For ax² + bc +c = 0, what do you find for 1 repeated root?

Answer 8

b² - 4ac = 0.

Tangent to the circle.

Question 9

For ax² + bc +c = 0, what do you find for 2 different roots?

Answer 9

b² - 4ac > 0.

Question 10

What do you need to consider to decide whether a point lies on, inside or outside a circle?

Answer 10

The distance of the point from the centre and the radius of the circle.

Question 11

For a Point (P) to lie on the circle, what must be true?

Answer 11

The distance of the point from the centre must be equal to the radius of the circle. (CP = r).

Question 12

For a Point (P) to lie inside the circle, what must be true?

Answer 12

The distance of the point from the centre must be less than the radius of the circle. (CP < r).

Question 13

For a Point (P) to lie outside the circle, what must be true?

Answer 13

The distance of the point from the centre must be more than the radius of the circle. (CP > r).

Question 14

What is the name of a line which is perpendicular to a tangent at the intersection of a tangent to a circle?

Answer 14

Normal line.

Question 15

What are the 4 Circle Geometry Theories?

Answer 15

• The centre is the midpoint of the Diameter.
• The angle in a semicircle is a right angle.
• If a line is drawn from the centre of a circle to a chord, at a right angle to that chord, it will bisect the chord and is the shortest distance from the chord to the centre.
• The tangent to a circle is perpendicular to the radius at that point and tangents from the same point (not on the circle) to a circle have the same length.

Question 16

What must be true for the Circle Geometry Theory involving the diameter?

Answer 16

The distance of the midpoint of the circle to the circumference is equal in any direction. (AC = BC = r).

Question 17

What must be true for the Circle Geometry Theory involving an angle in a semicircle?

Answer 17

It will always be a right angle no matter where the point is in a semicircle.

Question 18

What must be true for the Circle Geometry Theory involving a chord?

Answer 18

The distance of one point to the midpoint of the chord is equal either side and the perpendicular bisector meets at this midpoint. (AM = MB).

Question 19

What must be true for the Circle Geometry Theory involving Tangents?

Answer 19

The distance between the tangents and the point they meet the circle are equal where two tangents meet each other. (AP = BP).

Question 20

What must be true for the Circle Geometry Theory involving Tangents about the angles?

Answer 20

The angle at the centre is double that at the point where the 2 tangents meet.