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

A circle has equation

$x^2+y^2-10x+16y=80$

Find

(i) the coordinates of the centre of the circle,

(ii) the radius of the circle.

Given that $P$ is the point on the circle that is furthest away from the origin $O$, find the exact length $OP$

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

Find the equation of 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 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 $y=a+bx$.

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.

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 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 $y=mx$.

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

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

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.

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 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 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 $B$

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 $y=mx+c$.

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 circle $C$ has equation

$x^2+y^2-6x+10y+k=0$

where $k$ is a constant.

Find the coordinates of the centre of $C$.

Given that $C$ does not cut or touch the $x$-axis, find the range of possible values for $k$.

The circle $C$ has centre $A$ with coordinates $\left(7,5\right)$.

The line $l$, with equation $y=2x+1$, is the tangent to $C$ at the point $P$, as shown in Figure 3.

Show that an equation of the line $PA$ is $2y+x=17$.

Find an equation for $C$.

The line with equation $y=2x+k,k\ne 1$ is also a tangent to $C$.

Find the value of the constant $k$.

The circle $C$ has equation

$x^2+y^2-10x+4y+11=0$

Find

(i) the coordinates of the centre $C$,

(ii) the exact radius of $C$, giving your answer as a simplified surd.

The line $L$ has equation $y=3x+k$ where $k$ is a constant.

Given that $L$ is a tangent to $C$, find the possible values of $k$, giving your answers as simplified surds.

Figure 4 shows a sketch of a circle $C$ with centre $N(7,4)$

The line $l$ with equation $y=\frac{1}{3}x$ is a tangent to $C$ at the point $P$.

Find the equation of line $PN$ in the form $y=mx+c$, where $m$ and $c$ are constants,

Find an equation for $C$.

The line with equation $y=\frac{1}{3}x+k$, where $k$ is a non-zero constant, is also a tangent to $C$.

Find the value of $k$.

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 $C$has equation

$x^2+y^2+6kx-2ky+7=0$

where $k$ is a constant.

Find in terms of $k$,

(i) the coordinates of the centre of $C$

(ii) the radius of $C$

The line with equation$y=2x–1$ intersects $C$ at $2$ distinct points.

Find the range of possible values of $k$.

Figure 3 shows the circle $C$ with equation

$x^2+y^2-10x-8y+32=0$

and the line $l$ with equation

$2y+x+6=0$

Find

(i) the coordinates of the centre of $C$,

(ii) the radius of $C$

Find the shortest distance between $C$ and $l$.

A circle $C$ with radius $r$

• lies only in the 1st quadrant

• touches the $x$-axis and touches the $y$-axis

The line $l$ has equation $2x+y=12$

Show that the $x$ coordinates of the points of intersection of $l$ with $C$ satisfy

$5x^2+\left(2r-48\right)x+\left(r^2-24r+144\right)=0$

Given also that $l$ is a tangent to $C$, find the two possible values of $r$, giving your answers as fully simplified surds.

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 $C_2$.