Circles BARNES

Question 1

circle equation

Answer 1

( x - a )^2 + ( y - b )^2 = r^2
(a,b) is the centre

Question 2

To test if points are in, on, or outside a circle

Answer 2

compare the distance calculation to the radius

Question 3

A circle has radius 5, centre (-3, 6)
Write down the equation of the circle
Determine whether the points M(2, 6) and N(3, 1) lie in, on, or outside the circle

Answer 3

M is on
N is outside

Question 4

Sometimes you need to complete the square (twice) to

Answer 4

identify features in a circle equation

Question 5

Find the radius and coordinates of the centre of the circle
x^2 - 3x + y^2 + 4y = 12

Answer 5

centre = 3/2 , -2
radius = √73 / 2

Question 6

angle in a semicircle is

Answer 6

90°
circle theorem often shows up, such as in proving one side of a triangle is the diameter of a circle

Question 7

Points A(4, -5), B(2, 9) and C(9, 10) lie on a circle
Show that AC is a diameter of the circle
Hence find an equation of the circle

Answer 7

if AC is a diameter then ABC satisfies angles in a semicircle is 90 degrees and triangle ABC satisfies Pythagoras

equation is ( x - 13/2 )^2 + ( y - 5/2 )^2 = 250/4

Question 8

A tangent is

Answer 8

perpendicular to radius, passing through a single point on circumference

Question 9

A normal is

Answer 9

the radius to that point on the circumference

Question 10

to find normal or tangent

Answer 10

first find the gradient of the relevant radius

Question 11

Find the equation of the tangent to the circle (x - 3)^2 + (y + 5)2 = 5 at (2, -7)

Answer 11

y = -1/2x - 6

Question 12

Given equations for a line and circle how to find intersections

Answer 12

solve simultaneously

Question 13

d > r1 + r2

Answer 13

dont intersect (far away)

Question 14

r2 - r1 < d < r1 + r2

Answer 14

intersect at 2 points

Question 15

d < r2 - r1

Answer 15

dont intersect (in eachother)

Question 16

d = r1 + r2

Answer 16

intersect once (outside eachother)

Question 17

d = r2 - r1

Answer 17

intersect once (inside eachother)

Question 18

wo circles have equations x2 - 6x + y2 - 20y = -45 and (x - 15)2 + (y - 5)2 = r2
For the case r = 7, show the circles intersect at two different points
If instead the two circles are tangent to each other, find two possibles values of r

Answer 18

since 8-7 < 13 < 8+7, the circles overlap twice
if tangent, r2 = 5 or 21