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Higher Maths > Circle > Flashcards

Flashcards in Circle Deck (21)
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1
Q

the equation of a circle with centre (0,0) and radius r, is:

A

x squared + y squared = r squared

2
Q

how to work out radius if given a centre and a point the circle passes through

A

distance formula

3
Q

the equation of a circle with centre (a,b) and radius r is:

A

(x-a) squared + (y-b) squared = r squared

4
Q

how to work out circle equation if given the points of the diameter AB

A

find centre of circle (midpoint of AB)
use distance formula to work out radius using centre and either A or B
sub centre and radius into (x-a) squared + (y-b) squared = r squared

5
Q

given a circle with centre (a,b) and radius r units, how do you determine whether the point (p,q) lies within, outwith or on the circumference?

A

use (p-a) squared + (q-b) squared = r squared

r squared then outwith circle

6
Q

what is the general equation of a circle?

A

x squared + y squared + 2gx + 2fy + c = 0

7
Q

centre of circle from general equation

A

(-g,-f)

8
Q

radius of circle from general equation

A

square root of g squared + f squared - c

9
Q

what if g squared + f squared - c < 0

A

we cannot obtain a real value for the radius, since we would square a negative

not a circle

10
Q

what if g squared + f squared - c = 0

A

the radius is zero so the equation represents a point rather than a circle

11
Q

if a line and a circle only touch at one point, then the line is a…

A

tangent to the circle at that point

12
Q

to find out how many times a line and circle meet…

A

we can use substitution (sub line equation into circle equation)

then find y-coords by subbing x coords into easiest equation (normally line)

13
Q

to show that a line is a tangent to a circle…

A

the equation of the line can be substituted into the equation of the circle and solved - there should only be one solution

14
Q

2 methods to prove that a line is a tangent to a circle

A
  1. factorise - (since solutions are equal)
    * use method 1 if asks for point of contact/intersection*
  2. discriminant - b squared- 4ac=0
15
Q

if a line is a tangent to a circle, then the radius…

A

will meet the tangent at right angles

16
Q

steps to find equation of a tangent to a circle

A
  1. draw sketch
  2. find centre
  3. find gradient of radius
  4. find gradient of tangent using m1xm2=-1
  5. find equation of tangent using y-b=m(x-a)
17
Q

d > r1 + r2

A

the circles do not touch

18
Q

d = r1 + r2

A

the circles touch externally

19
Q

r1 - r2 < d > r1 + r2

A

the circles meet at two distinct points

20
Q

d = r1 - r2

A

the circles touch externally

21
Q

d < r1 - r2

A

the circles do not touch