Circle · Guided Practice

Circle

18 questions with answers and video solutions. Try each one before revealing the answer.

1
A is the point (−3, 8)\displaystyle (-3,\,8) and B is the point (1, −2)\displaystyle (1,\,-2). AB is the diameter of a circle.
Determine the equation of this circle.

No hints on this one — its full written solution is with the answer.

2
Circle C1\displaystyle C_1 has equation x2+y2+4x−6y−12=0\displaystyle x^2+y^2+4x-6y-12=0. Circle C2\displaystyle C_2 has centre (1,−4)\displaystyle (1,-4). The two circles have equal radii.
Find the equation of circle C2\displaystyle C_2.

No hints on this one — its full written solution is with the answer.

3
Circle C1\displaystyle C_1 has equation (x−1)2+(y+7)2=16\displaystyle (x-1)^2+(y+7)^2=16. Circle C2\displaystyle C_2 has equation x2+y2−12x−10y+12=0\displaystyle x^2+y^2-12x-10y+12=0.
(a) Write down the centres and radii of circles C1\displaystyle C_1 and C2\displaystyle C_2.
(b) Show that C1\displaystyle C_1 and C2\displaystyle C_2 do not intersect.

No hints on this one — its full written solution is with the answer.

4
The point P(6,−4)\displaystyle P(6,-4) lies on the circle x2+y2−4x+2y−20=0\displaystyle x^2+y^2-4x+2y-20=0.
Find the equation of the tangent to the circle at P.

No hints on this one — its full written solution is with the answer.

5
Show that the line y=2x−3\displaystyle y=2x-3 is a tangent to the circle x2+y2−8x+11=0\displaystyle x^2+y^2-8x+11=0.
Find the coordinates of the point of contact.

No hints on this one — its full written solution is with the answer.

6
The circle x2+y2−6x−14y+k=0\displaystyle x^2+y^2-6x-14y+k=0 is a tangent to one of the coordinate axes and intersects the other axis at two distinct points.
Determine the value of k\displaystyle k.

No hints on this one — its full written solution is with the answer.

7
x2+y2−2px−4py+3p+2=0\displaystyle x^2+y^2-2px-4py+3p+2=0 is the equation of a circle.
Determine the range of values of p\displaystyle p.

No hints on this one — its full written solution is with the answer.

8
2016 P1 Q8
5 Marks
Show that the line with equation y=3x−5\displaystyle y=3x-5 is a tangent to the circle with equation
x2+y2+2x−4y−5=0\displaystyle x^{2}+y^{2}+2x-4y-5=0
and find the coordinates of the point of contact.
9
2017 P1 Q2
4 Marks
The point P(−2,1)\displaystyle P(-2,1) lies on the circle
x2+y2−8x−6y−15=0\displaystyle x^{2}+y^{2}-8x-6y-15=0
Find the equation of the tangent to the circle at P.
10
2019 P1 Q3
2 Marks
Circle C1\displaystyle C_{1} has equation
x2+y2−6x−2y−26=0\displaystyle x^{2}+y^{2}-6x-2y-26=0
Circle C2\displaystyle C_{2} has centre (4,−2)\displaystyle (4,-2).
The radius of C2\displaystyle C_{2} is equal to the radius of C1\displaystyle C_{1}.
Find the equation of circle C2\displaystyle C_{2}.
11
2021 P1 Q15
ABCD is a square containing four congruent circles. A is the point (2, 1)\displaystyle (2,\,1), and D is the point (10, 1)\displaystyle (10,\,1).
Determine the equation of the circle with centre P (the top-right circle).

No hints on this one — its full written solution is with the answer.

12
2023 P2 Q15
4 Marks

The line x+3y=17\displaystyle x+3y=17 is a tangent to a circle at the point (2, 5).

Circle tangent diagram

The centre of the circle lies on the y-axis.
Find the coordinates of the centre of the circle.

13
2024 P1 Q7
4 Marks

The line y=2x\displaystyle y=2x is a tangent to the circle with equation
x2+y2−14x−8y+45=0\displaystyle x^{2}+y^{2}-14x-8y+45=0
Determine the coordinates of the point of contact.

14
2024 P2 Q10
(2, 2)4 Marks

The circle C1\displaystyle C_{1} has equation x2+y2+18x−2y−8=0\displaystyle x^{2}+y^{2}+18x-2y-8=0.
(a)  Find the centre and radius of C1.\displaystyle C_{1}.
A second circle, C2,\displaystyle C_{2}, touches C1\displaystyle C_{1} internally.
The centre of C2\displaystyle C_{2} is (-6, 0).

Two circles touching internally

(b)  Determine the equation of C2\displaystyle C_{2}.

15
2025 P1 Q9
4 Marks

Find the coordinates of the points of intersection of the line with equation y=x+1\displaystyle y=x+1 and the circle with equation
x2+y2−2x+6y−15=0\displaystyle x^{2}+y^{2}-2x+6y-15=0

16
2025 P2 Q14
(2, 2, 3)7 Marks

Circle C1\displaystyle C_{1} has equation (x+5)2+(y−6)2=9.\displaystyle (x+5)^{2}+(y-6)^{2}=9.
(a)  State the centre and radius of C1.\displaystyle C_{1}.
Circle C2\displaystyle C_{2} has equation x2+y2−14x+6y+54=0.\displaystyle x^{2}+y^{2}-14x+6y+54=0.
(b)  State the centre and radius of C2.\displaystyle C_{2}.
Circles C1\displaystyle C_{1}, C2\displaystyle C_{2} and C3\displaystyle C_{3} are touching as shown in the diagram. The centre of circle C3\displaystyle C_{3} lies on the line joining the centres of C1\displaystyle C_{1} and C2.\displaystyle C_{2}.

Circles C1, C2 and C3

(c)  Determine the equation of C3.\displaystyle C_{3}.

17
2026 P1 Q3
4 Marks

A circle has equation x2+y2+2x−4y−20=0.\displaystyle x^{2}+y^{2}+2x-4y-20=0.
Point A(2,6) lies on the circle.

Circle with centre marked and the tangent drawn at point A on the circumference

Find the equation of the tangent to the circle at A.

18
2026 P2 Q4
2 Marks
A circle has centre (−2,5).\displaystyle (-2,5).
The point (3,7)\displaystyle (3,7) lies on the circle.
Find the equation of the circle.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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