Edexcel A Level Maths
Topic questions Pure mathematics Equation of a circle
18 questions · 149 marks · about 3 hours
Also involves Surds
The circle C has equation
State the coordinates of the centre of C.
Find the radius of C writing your answer as a fully simplified surd.
Determine, giving a reason, whether or not the origin lies inside the circle C.
The circle C1 has equation
Find
the coordinates of the centre of C1
the radius of C1
A different circle C2
• has centre with coordinates (–6, –8)
• has radius k, where k is a constant
Given that C1 and C2 intersect at 2 distinct points,
find the range of values of k, writing your answer in set notation.
A circle C has equation
where k is a constant.
Find in terms of k,
the coordinates of the centre of C
the radius of C
The line with equation y = 2x – 1 intersects C at 2 distinct points.
Find the range of possible values of k.
The circle C has equation
Find
the coordinates of the centre of C,
the exact radius of C, giving your answer as a simplified surd.
The line l has equation y = 3x + k where k is a constant.
Given that l is a tangent to C,
find the possible values of k, giving your answers as simplified surds.
Also involves Surds
The circle C has equation
Find
the coordinates of the centre of C,
the exact radius of C, giving your answer as a simplified surd. The line l has equation y = 3x + k where k is a constant. Given that l is a tangent to C,
find the possible values of k, giving your answers as simplified surds.
Also involves Sine and cosine rules · Arc length and sector area
Circle C1 has equation x2 + y2 = 100 Circle C2 has equation (x − 15)2 + y2 = 40 The circles meet at points A and B as shown in Figure 3.
Show that angle AOB = 0.635 radians to 3 significant figures, where O is the origin. The region shown shaded in Figure 3 is bounded by C1 and C2
Find the perimeter of the shaded region, giving your answer to one decimal place.
Also involves Surds · The discriminant
A circle C with radius r ● lies only in the 1st quadrant ● touches the x-axis and touches the y-axis The line l has equation 2x + y = 12
Show that the x coordinates of the points of intersection of l with C satisfy
Given also that l is a tangent to C,
find the two possible values of r, giving your answers as fully simplified surds.
Also involves Parallel and perpendicular lines · Tangent to a circle
The circle C has centre A with coordinates (7, 5).
The line l, with equation y 
2x 
1, is the tangent to C at the point P, as shown in Figure 3.
Show that an equation of the line PA is 2y 
x 
17
Find an equation for C.
The line with equation y 
2x + k, k 
1 is also a tangent to C.
Find the value of the constant k.
Also involves Completing the square · Straight line graphs
The circle C has equation
Find
the coordinates of the centre of C,
the exact value of the radius of C,
the y coordinates of the points where the circle C crosses the y-axis.
Find an equation of the tangent to C at the point (2, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers.
Also involves Surds · Completing the square
The circle C has equation
Find
the coordinates of the centre of C,
the radius of C,
the y coordinates of the points where the circle C crosses the line with equation x = 4, giving your answers as simplified surds.
Also involves Coordinate geometry · Parallel and perpendicular lines
The circle C has centre P (7, 8) and passes through the point Q (10, 13), as shown in Figure 2.
Find the length PQ, giving your answer as an exact value.
Hence write down an equation for C.
The line l is a tangent to C at the point Q, as shown in Figure 2.
Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers.
Also involves Surds
Figure 3 shows a circle C with centre Q and radius 4 and the point T which lies on C. The tangent to C at the point T passes through the origin O and OT = 6 ¥ 5 Given that the coordinates of Q are (11, k), where k is a positive constant,
find the exact value of k,
find an equation for C.
The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4.
Write down an equation for the circle C, that is shown in Figure 4.
A line through the point P (8, – 7) is a tangent to the circle C at the point T.
Find the length of PT.
Also involves Coordinate geometry
The circle C has equation
The centre of C is at the point M.
Find
the coordinates of the point M,
the radius of the circle C.
N is the point with coordinates (25, 32).
Find the length of the line MN.
The tangent to C at a point P on the circle passes through point N.
Find the length of the line NP.
The circle C with centre T and radius r has equation
Find the coordinates of the centre of C.
Show that r = 5
The line L has equation x = 13 and crosses C at the points P and Q as shown in Figure 1.
Find the y coordinate of P and the y coordinate of Q.
Given that, to 3 decimal places, the angle PTQ is 1.855 radians,
find the perimeter of the sector PTQ.
Also involves Surds · Quadratic equations
The circle C has equation
Find
the coordinates of the centre of C,
the radius of C,
the coordinates of the points where C crosses the y-axis, giving your answers as simplified surds.
Also involves Coordinate geometry
The circle C has equation
Find the centre and the radius of C.
The point P (–1, 1) and the point Q (7, –5) both lie on C.
Show that PQ is a diameter of C.
The point R lies on the positive y-axis and the angle PRQ = 90°.
Find the coordinates of R.
Also involves Coordinate geometry · Parallel and perpendicular lines
The circle C has centre (3, 1) and passes through the point P (8, 3).
Find an equation for C.
Find an equation for the tangent to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.
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