1. Write down the equation for each circle:
|
a
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Centre (0. 0), radius 5 |
b
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Centre (0. 0), radius 3 |
|
c
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Centre (2, 1), radius 5 |
d
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Centre (7, 3), radius 4 |
|
e
|
Centre (-3, 2), radius 2 |
f
|
Centre (5, -1), radius 10 |
|
g
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Centre (3. 0), radius 3 |
h
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Centre (0. -2), radius 6 |
2. Give the centre and radius for each circle:
|
a
|
x2 + y2 = 81 |
b
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y2 = 121 − x2 |
|
c
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(x − 2)2 + (y − 3)2 = 16 |
d
|
(x − 5)2 + (y − 4)2 = 9 |
|
e
|
(x + 1)2 + (y − 6)2 = 25 |
f
|
(x + 3)2 + (y +1)2 = 49 |
|
g
|
(x + 7)2 + y2 = 64 |
h
|
x2 + (y − 4)2 = 16 |
3. By completing the square, find the centre and radius for each circle:
x2 + 2x + y2 + 6y = 15 x2 + 8x + y2 + 10y = 8 x2 − 10x + y2 − 8y = 8 x2 − 6x + y2 + 6y = 7 x2 + 12x + y2 − 2y − 12 = 0 x2 − 6x + y2 + 8y = 0 x2 − 4x + y2 − 4y + 7 = 0 x2 + 5x + y2 − 3y = 0
4. Sketch the graphs of each ellipse:
|
a
|
 |
b
|
 |
|
c
|
 |
d
|
 |
5. a. Sketch a graph to show x2 + y2 = 100
b. Mark the points A = (6, 8) and B = (8, 6)
c. Find the gradient AB.
d. Find the mid-point of AB.
e. Find the gradient perpendicular to AB.
f. Hence find the equation of the perpendicular bisector of AB.
g. Show this line passes through the centre of the circle.
h. What property of the circle does this demonstrate?
6. a. Sketch a graph to show x2 + y2 = 25
b. Mark the points C = (3, 4) and D = (-4, 3)
c. Find the equation of the line CD.
d. Find the coordinates of M, the mid-point of CD.
e. Give the coordinates of the centre of the circle O.
f. Give the equation of the line OM.
g. Prove that the lines OM and CD are perpendicular.
h. What property of the circle does this demonstrate?
7. a. Sketch a graph to show x2 − 6x + y2 = 0
b. Find the two points, A and B where this circle crosses the x-axis.
c. Prove that AB is a diameter of the circle.
d. Show that the point C = (4, √8) lies on the circle.
e. Find the gradients of AC and of BC.
f. What property of the circle can you deduce from this?
8. a. Sketch the graph of x2 + 12x + y2 + 11 = 0
b. Find the coordinates of the centre point, O
c. Prove that the point P ( -3, 4) is on the circle.
d. Find the gradient of OP.
e. Let Q be a variable point (x0, 0). Find the gradient of PQ.
f. Choose a value for x0 so that PQ is perpendicular to OP.
g. What is now special about the line PQ.
h. Find the lengths of the lines OP, PQ and OQ.
i. What property of the circle does this demonstrate?
