Circles (AQA AS Maths: Pure): Exam Questions

A circle has centre (6, -5) and goes through the point (1, 7).  Find the equation of the circle.

Show that can be written in the form  , where ,  and are integers to be found.

Hence write down the centre and radius of the circle with equation

The linemeets the circle with equation .

(i) Show that the line and circle meet at one point only.

(ii) Find the coordinates of the point of intersection.

The lineintersects the circle at the points and . Find the coordinates of and .

A circle  has centre (-4, 1) and passes through the point (0, 3). 

Find an equation for the circle .

Find an equation for the tangent to the circle at .

The points and lie on a circle.

Show that triangle is a right-angle triangle.

Explain why the line segment  must be the diameter of the circle.

Hence find the equation of the circle.

Circles ,  and all have their centres on the -axis.  

Circle  has equation .  

Circle  has equation .  

Circles  and touch at point , and circles  and  touch at point .

Find the coordinates of the centre of circle .

A circle has equation .

The lines  and  are both tangents to the circle, and they intersect at the origin.

Explain why the equations for  and  must each be in the form , where  is the gradient of the line.

Show that the gradients of and  must be the solutions to the equation

Hence find the equations of  and  , giving your answers in the form .

The points A (-3, 1) and B (3, -7) are the two endpoints of the diameter AB of a circle. Find the equation of the circle.

Show that can be written in the form , where and are constants to be found.

Hence write down the centre and radius of the circle with equation

The line meets the circle with equation .

(i) Show that the line and circle meet at one point only.

(ii) Find the coordinates of the point of intersection.

The line intersects the circle at the points and .  Find the coordinates of and

A circle  has centre (-2, 3) and passes through the point (6, -3). 

Find an equation for the circle .

Find an equation for the tangent to the circle at P.

The points and lie on a circle.

Show that  

Deduce a geometrical property of the line segment AB.

Hence find the equation of the circle.

Triangle  has vertices (-8, 1), (12, 16) and (12, 1).  A circle with equation touches Triangle  at the three points and , as shown in the diagram below:

Write down the coordinates of points R and Q.

Find the coordinates of point P.

A circle has equation

The lines and  are both tangents to the circle, and they intersect at the point (0, 14).

Find the equations of  and , giving your answers in the form .

The points (2, -21) and (-5, 3) are the two endpoints of the diameter of a circle. Find the equation of the circle in the form , where and  are integers to be found.

Find the centre and radius of the circle with equation

The line intersects the circle at exactly two points.  Find the range of possible values of .

The points (-2, 3), (0, 6) and (, -1) lie on a circle, where  is the diameter of the circle.

Find the value of .

A circle has equation Point lies on the circle, and the tangent to the circle at point  has a gradient of -3. Find the two possible sets of coordinates for point .

The pointsand lie on a circle.

Find the equation of the circle.

A circle has equation

The lines and  are both tangents to the circle, and they intersect at the point (5, 0).

Find the equations of  and , giving your answers in the form .

The diagram below shows circles  and which intersect at the two points  and . Circle  has equation  , and points  and lie along the line with equation  . Circle also passes through the point (-13, 2). 

Find an equation of circle .