Alevel_C1C2

4. (a) Centre (-3, 3), radius 3 A2 (2)
(b)
Substitute y = x + 3 in the equation of the circle:
(x + 3) 2 + x 2 = 9 B1
2x 2 + 6x = 0
B1
x(x + 3) = 0
B1
x = -3, then y = 0; A(-3, 0)
A1ft
x = 0, then y = 3; B(0, 3) A1ft (4)
(c) (x + 3) 2 + (x + a - 3) 2 = 9 M1
x 2 + 6x + 9 + x 2 + 2(a - 3) + (a - 3) 2 = 9
2x 2 + 2ax + (a - 3) 2 = 0
B1oe
Since the line touches the circle, therefore the discriminant = 0
( 2a) − 4 × 2 × ( a − 3)
= 0
M1
a 2 - 12a + 18 = 0
(a - 6) 2 = 18, then a = 6 ± 3 2 M1oe A2 (6)
12
5. (a) Gradient of AC is − 2 3
A1
y − 3 =
2
− ( x + 2); 3y + 2x
= 5
A1 (2)
3
(b) Since C is the midpoint of AB, then B1
− 2 + x
= 4
2
M1oe
Therefore the x-coordinate of B is 10.
A1
Substitute x = 10 in 3y + 2x = 5:
y = 5 A1 (4)
6
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Core C2