Circles (Cambridge (CIE) AS Maths: Pure 1): Exam Questions

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

Show that $x^2+y^2+2x-6y+9=0$ can be written in the form  ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$, where $a$, $b$ and $r$ are integers to be found.

Hence write down the centre and radius of the circle with equation $x^2+y^2+2x-6y+9=0.$

The line$x+y=-7$meets the circle with equation ${(x-1)}^2+{(y-2)}^2=50$.

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

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

The line$7x+y=-6$intersects the circle ${\left(x-2\right)}^2+{\left(y-5\right)}^2=25$at the points $A$and $B$. Find the coordinates of $A$and $B$.

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

Find an equation for the circle $C$.

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

The points $A(3,5),B(5,3)$ and $C(9,7)$ lie on a circle.

Show that triangle $ABC$ is a right-angle triangle.

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

Hence find the equation of the circle.

Circles $C_1$, $C_2$ and $C_3$ all have their centres on the $x$-axis.  

Circle $C_1$ has equation ${\left(x+7\right)}^2+y^2=4$.  

Circle $C_3$ has equation $x^2+y^2-10x+16=0$.  

Circles $C_1$ and $C_2$ touch at point $A$, and circles $C_2$ and $C_3$ touch at point $B$.

Find the coordinates of the centre of circle $C_2$.

A circle has equation $x^2+y^2-12x+14y=-68$.

The lines $l_1$ and $l_2$ are both tangents to the circle, and they intersect at the origin.

Explain why the equations for $l_1$ and $l_2$ must each be in the form $y=mx$, where $m$ is the gradient of the line.

Show that the gradients of $l_1$ and $l_2$ must be the solutions to the equation

$19m^2+84m+32=0.$

Hence find the equations of  $l_1$ and  $l_2$, giving your answers in the form $y=mx.$.

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 $x^2+y^2+5x-2y-5=0$ can be written in the form ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$, where $a,b$ and $r$ are constants to be found.

Hence write down the centre and radius of the circle with equation $x^2+y^2+5x-2y-5=0.$

The line $y+2x=11$meets the circle with equation $x^2+y^2+6x-14y=-38$ .

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

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

The line $x+5y+22=0$ intersects the circle $x^2+y^2+4x+8y-6=0$at the points $A$ and $B$. 
Find the coordinates of $A$and $B$

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

Find an equation for the circle $C$.

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

The points $A(-3,6),B(5,-4)$ and $C(6,5)$ lie on a circle.

Show that $\angle ACB=90°.$ 

Deduce a geometrical property of the line segment AB.

Hence find the equation of the circle.

Triangle $ABC$ has vertices $A$(-8, 1), $B$(12, 16) and $C$(12, 1).  A circle with equation ${\left(x-7\right)}^2+{\left(y-6\right)}^2=25$ touches Triangle $ABC$ at the three points $P,Q$and $R$, 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 $x^2+y^2+14x-6y=-41.$

The lines $l_1$and $l_2$ are both tangents to the circle, and they intersect at the point (0, 14).

Find the equations of $l_1$ and $l_2$, giving your answers in the form $y=mx+c$.

The points $A$(2, -21) and $B$(-5, 3) are the two endpoints of the diameter $AB$of a circle.
Find the equation of the circle in the form $a{x}^2+a{y}^2+bx+cy+d=0$, where $a,b,c$ and $d$ are integers to be found.

Find the centre and radius of the circle with equation $x^2+y^2+x-3y+2=0.$

The line $x+y=c$ intersects the circle $x^2+y^2-6x+10y-16=0$ at exactly two points.  Find the range of possible values of $c$.

The points $A$(-2, 3), $B$(0, 6) and $C$($k$, -1) lie on a circle, where $BC$ is the diameter of the circle.

Find the value of $k$.

A circle $C$ has equation $x^2+y^2-10x-4y+19=0.$ Point $P$ lies on the circle, and the tangent to the circle at point $P$ has a gradient of -3. Find the two possible sets of coordinates for point $P$.

The points$A(4,6),B(7,2)$and $C(12,12)$ lie on a circle.

Find the equation of the circle.

A circle has equation $x^2+y^2+4x+12y=-23.$

The lines $l_1$ and $l_2$ are both tangents to the circle, and they intersect at the point (5, 0).

Find the equations of $l_1$ and $l_2$, giving your answers in the form $y=mx+c$.

The diagram below shows circles $C_1$ and $C_2$ which intersect at the two points $A$ and $B$.
Circle $C_1$ has equation  $x^2+y^2-16x-10y+39=0$, and points $A$ and $B$ lie along the line with equation  $3x-y=-1$. Circle $C_2$also passes through the point (-13, 2). 

Find an equation of circle $C_2$.