The circle C has equation (x − 3)2 + (y + 5)2 = 20
Write down the coordinates of the centre of C.
Find the radius of C, giving your answer as a simplified surd.
2 marks
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a) (3, -5)b) 2√5
2
The circle C has equation x2 + y2 − 8x + 6y + 9 = 0. Find the coordinates of the centre of C and the radius of C.
3 marks
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centre (4, -3), radius 4
3
A circle has centre (−2, 1) and passes through the point (1, 5). Find an equation for the circle.
3 marks
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(x + 2)2 + (y - 1)2 = 25
4
The points A(−1, 4) and B(5, −2) are the ends of a diameter of the circle C.
Find an equation for C.
Determine, giving a reason, whether the origin lies inside or outside C.
5 marks
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a) (x - 2)2 + (y - 1)2 = 18b) inside, because its distance from the centre is √5, less than √18
5
The circle C has equation (x − 1)2 + (y + 2)2 = 25. The point P(4, 2) lies on C. Find an equation of the tangent to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.
4 marks
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3x + 4y - 20 = 0
6
The circle C has equation x2 + y2 − 6x + 4y + 3 = 0. The line l with equation y = x − 1 meets C at the points A and B.
Find the coordinates of A and B.
Find the exact length of the chord AB.
Show that the perpendicular bisector of AB passes through the centre of C.
7 marks
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a) (0, -1) and (2, 1)b) 2√2c) shown: y = 1 - x passes through (3, -2)
7
The circle C has equation (x − 1)2 + (y − 2)2 = 5. The line l has equation y = 2x + k, where k is a constant.
Show that the x coordinates of any points where l meets C satisfy 5x2 + (4k − 10)x + k2 − 4k = 0
Given that l is a tangent to C, find the possible values of k.
6 marks
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a) shownb) k = 5 or k = -5
8
The points P(−2, 4), Q(4, 6) and R(6, 0) lie on the circle C.
Show that angle PQR is a right angle.
Hence find an equation for C.
6 marks
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a) shown: gradients 1/3 and -3 multiply to -1b) (x - 2)2 + (y - 2)2 = 20
9
The points A(7, 5), B(−1, 5) and D(5, −1) lie on the circle C.
Find the equation of the perpendicular bisector of AD.
Hence, or otherwise, find the coordinates of the centre of C.
Find an equation for C.
7 marks
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a) y = 4 - x/3, or x + 3y - 12 = 0b) (3, 3)c) (x - 3)2 + (y - 3)2 = 20
10
The circle C has equation x2 + y2 − 4x + 2y − 4 = 0 and centre M. The point P has coordinates (8, 7). A line from P touches C at the point T.
Find the centre and radius of C.
Find the exact length of PT.
Find the size of angle MPT, giving your answer in degrees to 1 decimal place.
5 marks
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a) centre (2, -1), radius 3b) √91c) 17.5°
11
The equation x2 + y2 − 2kx + 4y + 2k + 4 = 0, where k is a constant, represents a circle C.
Find, in terms of k, the coordinates of the centre of C and an expression for the square of its radius.
Hence find the set of possible values of k.
Given that C touches the x-axis, find the exact possible values of k.
7 marks
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a) centre (k, -2), r2 = k2 - 2kb) k 2c) k = 1 + √5 or k = 1 - √5
12
A mobile phone mast in rural Cumbria has coordinates (3, 4) relative to a fixed origin O, where the units are kilometres. The mast gives a signal everywhere within 5 km of it, and the boundary of this region is modelled as a circle.
A straight road through a village at O is modelled by the line with equation y = 12x
Write down an equation for the boundary circle.
Find the coordinates of the points where the road meets the boundary.
Find the exact length of road that has a signal.
A van travels along the road at a constant 60 km/h. Find, to the nearest minute, how long the van has a signal.
Give one reason why the circle may not be a good model for the region with a signal.
8 marks
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a) (x - 3)2 + (y - 4)2 = 25b) (0, 0) and (8, 4)c) 4√5 kmd) 9 minutese) for example: hills or buildings block the signal, so the edge is not a perfect circle
Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.