# **ANSWERS** - AS Maths - Core 1 - Solomon Answers - Circles - B

Answers to practice worksheets for Core 1 - topic cirles

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C1 CIRCLES Answers - Worksheet B
1 a (x - 3)2 + (y + 2)2 = 25 2 a = ( -52+ 3 , 6+8
2
) = (-1, 7)
2 2
b sub. (x - 3) + [(2x - 3) + 2] = 25 b radius = 16 + 1 = 17
(x - 3)2 + (2x - 1)2 = 25 (x + 1)2 + (y - 7)2 = 17
7-6
x2 - 2x - 3 = 0 c grad of radius = = 1
4
-1 - (-5)
(x + 1)(x - 3) = 0 grad of tangent = -4
x = -1, 3 y - 6 = -4(x + 5)
(-1, -5) and (3, 3) [ y = -4x - 14 ]
AB2 = 42 + 82 = 80
AB = 80 = 16 × 5 = 4 5
3 a (x + 4)2 - 16 + (y - 8)2 - 64 + 62 = 0 4 a PQ = 1 + 9 = 10
2 2
(x + 4) + (y - 8) = 18 radius = 1
2
PQ = 1
2
10
centre (-4, 8) radius 3 2 b = midpoint of PR
0+7 4+3
b grad of l = 2 grad of perp. = - 1
2
=( 2 , 2 )=(7 2
, 7
2
)
0 +1 +1
eqn. of line perp to l through centre: c midpoint of PQ = ( 2
, 42 )=(1
2
, 5
2
)
y - 8 = - (x + 4)1
2
centre of C1 = midpoint of ( , 1
2
5
2
) and ( 7
2
, 7
2
)
1+7 5+7
y=6- 1
2
x =( 2 2
2
, 2 2
2
) = (2, 3)
intersects l when: eqn. of C1:
2x + 1 = 6 - 12
x (x - 2)2 + (y - 3)2 = ( 1
2
10 )2
x = 2 (2, 5) is closest point x2 - 4x + 4 + y2 - 6y + 9 = 5
2
dist. (2, 5) to centre 2x2 - 8x + 8 + 2y2 - 12y + 18 = 5
= 36 + 9 = 45 = 3 5 2x2 + 2y2 - 8x - 12y + 21 = 0
min. dist. = 3 5 - 3 2 = 3( 5 - 2)
5 a midpoint AB = ( 0 +
2
2
, 3+ 7
2
) = (1, 5) 6 AP2 = (x + 3)2 + (y - 4)2
7-3
grad AB = =2 BP2 = x2 + (y + 2)2
2-0
2
AP = 2BP AP2 = 4BP2
y-5= -1
2
(x - 1) (x + 3)2 + (y - 4)2 = 4[x2 + (y + 2)2]
[y= 11
2
- 1
2
x] x2 + 6x + 9 + y2 - 8y + 16 = 4x2 + 4y2 + 16y + 16
b circle touches y-axis at (0, 3) x2 - 2x + y2 + 8y - 3 = 0
y-coord of centre = 3 (x - 1)2 - 1 + (y + 4)2 - 16 - 3 = 0
sub. 3 = 11 2
- 12
x (x - 1)2 + (y + 4)2 = 20
x=5 in form (x - a)2 + (y - b)2 = r2 circle
(x - 5)2 + (y - 3)2 = 25
7-3
3
2-5
4
y-7= 3
4
(x - 2)
4y - 28 = 3x - 6
3x - 4y + 22 = 0
Solomon Press

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C1 CIRCLES Answers - Worksheet B page 2
7 a = ( -4 +2( -2) , 9 + ( -5)
2
) = (-3, 2) 8 a x2 + (y - 2)2 - 4 - 16 = 0
b radius = 1 + 49 = 50 centre (0, 2)
(x + 3)2 + (y - 2)2 = 50 b C2: (x - 1)2 - 1 + (y - 4)2 - 16 - 60 = 0