Circles (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

A circle has centre (6, −5) and passes 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 $(x-a{)}^2+(y-b{)}^2=r^2$, where $a$, $b$ and $r$ are integers to be found.

Hence write down the centre and the 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 the circle meet at exactly one point.

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

The line $7x+y=-6$ intersects the circle $(x-2{)}^2+(y-5{)}^2=25$ 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-angled 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 $(x+7{)}^2+y^2=4$.

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 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 $B$.

The line with equation $y=x-1$ intersects the circle with equation $(x-5{)}^2+(y-4{)}^2=18$ at two distinct points.

Find the coordinates of the two points of intersection.

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 $(x-a{)}^2+(y-b{)}^2=r^2$, where $a$, $b$ and $r$ are constants to be found.

Hence write down the centre and the 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 the circle meet at exactly one point.

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

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 $(x-7{)}^2+(y-6{)}^2=25$ touches triangle $ABC$ at the three points $P$, $Q$ and $R$, as shown in the diagram below.

Write down the coordinates of the points $R$ and $Q$.

Find the coordinates of the 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 $y=mx+c$.

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

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 $k$.

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.

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 $c$.

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 $C_2$.