Circles - [Pure 3]. Flashcards
How can you best describe a circle?
A series of points which are equidistant from a fixed centre.
What is the General Form Equation of a Circle?
(x - a)2 + (y - b)2 = r2.
What is the Transformation from x2 + y2 = r2, to (x - a)2 + (y - b)2 = r2?
A Translation by vector [a b].
How do you find the centre and radius of a circle from an equation in the form: x2 + y2 + 2ax + 2by + c = 0?
Complete the Square.
* ((x + a)2 - (2a)2) + ((y + b)2 - (2b)2) + c = 0.
goes to…
- (x + a)2 + (y + b)2 = d.
When a line and a circle meet how many different roots can there be?
0 (no roots - don’t intersect), 1 (repeated root) or 2 (different roots).
What can you use to work out how many times a line and a circle intersect?
b2 - 4ac.
For ax2 + bc +c = 0, what do you find for no roots?
b2 - 4ac < 0.
For ax2 + bc +c = 0, what do you find for 1 repeated root?
b2 - 4ac = 0.
Tangent to the circle.
For ax2 + bc +c = 0, what do you find for 2 different roots?
b2 - 4ac > 0.
What do you need to consider to decide whether a point lies on, inside or outside a circle?
The distance of the point from the centre and the radius of the circle.
For a Point (P) to lie on the circle, what must be true?
The distance of the point from the centre must be equal to the radius of the circle. (CP = r).
For a Point (P) to lie inside the circle, what must be true?
The distance of the point from the centre must be less than the radius of the circle. (CP < r).
For a Point (P) to lie outside the circle, what must be true?
The distance of the point from the centre must be more than the radius of the circle. (CP > r).
What is the name of a line which is perpendicular to a tangent at the intersection of a tangent to a circle?
Normal line.
What are the 4 Circle Geometry Theories?
- 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.