Circles (AQA A Level Maths: Pure): Exam Questions

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 this point.

Express

$x^2+y^2+5x-2y-5=0$

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 find the radius and the coordinates of the centre of the circle with equation

$x^2+y^2+5x-2y-5=0$

A circle has centre $\left(6,-5\right)$ and passes through the point $\left(1,7\right)$. 

Find the equation of the circle.

The points $A\left(-3,1\right)$ and $B\left(3,-7\right)$ are the endpoints of the diameter $AB$ of a circle.

Find an equation for the circle.

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 this point.

A circle $C$ has equation

$x^2+y^2-12x+10y+k=0$

where $k$ is a constant.

Find the coordinates of the centre of $C$.

State the range of possible values for $k$.

Given that $k=36$, write down the radius of $C$.

A circle $C$ has a centre with coordinates $\left(-4,1\right)$ and passes through the point $P$ with coordinates $\left(0,3\right)$.

Find an equation for $C$.

Find the equation of the tangent to the circle at $P$.

Give your answer in the form $y=a+bx$.

A circle $C$ has centre $\left(1,-2\right)$ and passes through the point $P\left(5,1\right)$.

Find the equation of the circle.

Find an equation to the tangent of the circle at point $P$.

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.

Find the equation of the circle.

The straight line $x+5y+22=0$ intersects the circle with equation

$x^2+y^2+4x+8y-6=0$

at the points $A$ and $B$. 

Find the coordinates of $A$and $B$

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 $A$

• Circles $C_2$ and $C_3$ touch at point $B$.

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

A circle $C$ has a centre with coordinates $\left(-2,3\right)$ and passes through the point $P$ with coordinates $\left(6,-3\right)$.

Find an equation for $C$.

Find the equation of the tangent to $C$ at $P$.

Give your answer in the form $y=mx+c$.

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 have the form $y=mx$.

Show that $m$ satisfies the equation

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

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

Find the exact radius and the coordinates of the centre of the circle with equation

$x^2+y^2+x-3y+2=0$

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

Find the value of $k$.

A circle $C$ has equation

$x^2+y^2+4kx-8ky+5=0$

where $k$ is a constant.

The line with equation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«semantics»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«annotation encoding=¨application/vnd.wiris.mtweb-params+json¨»{¨fontFamily¨:¨Times New Roman¨,¨fontSize¨:¨18¨,¨autoformat¨:true,¨toolbar¨:¨«toolbar ref=`general`»«tab ref=`general`»«removeItem ref=`setColor`/»«removeItem ref=`bold`/»«removeItem ref=`italic`/»«removeItem ref=`autoItalic`/»«removeItem ref=`setUnicode`/»«removeItem ref=`mtext` /»«removeItem ref=`rtl`/»«removeItem ref=`forceLigature`/»«removeItem ref=`setFontFamily` /»«removeItem ref=`setFontSize`/»«/tab»«/toolbar»¨}«/annotation»«/semantics»«/math» intersects $C$ at two distinct points. Find the range of possible values for $k$.

The straight 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 $A\left(2,-21\right)$ and $B\left(-5,3\right)$ are the two endpoints of the diameter $AB$of a circle.

Find the equation of the circle in the form

$x^2+y^2+ax+by+c=0$

where $a,b$ and $c$ are integers to be found.

Triangle $ABC$ has vertices $A\left(-8,1\right)$, $B\left(12,16\right)$ and $C\left(12,1\right)$.

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.

Find the coordinates of points $R$ and $Q$.

Find the coordinates of point $P$.

A circle $C$ has equation

$x^2+y^2-10x-4y+19=0$

Point $P$ lies on $C$, and the tangent to $C$ at $P$ has a gradient of $-3$.

Find the two possible sets of coordinates for 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 $\left(0,14\right)$.

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

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 $\left(5,0\right)$.

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 $C_2$ intersecting at the points $A$ and $B$.

• Circle $C_1$ has equation  $x^2+y^2-16x-10y+39=0$

• Points $A$ and $B$ lie along the line with equation  $3x-y=-1$

• Circle $C_2$passes through the point $\left(-13,2\right)$

Find an equation of circle $C_2$.