Circles (Edexcel IAL Maths): Exam Questions

Write down the equations of the circles with the following centres and radii

(i) Centre: (0 , 0) Radius: *r *= 4,

(ii) Centre: (3 , -4) Radius: *r *= 2,

(iii) Centre: (-5 , 0) Radius: *r *= 5.

Write down the centre and the radius for each of the following circles

(i) $x^{2}+y^{2}=5^{2}$,

(ii) $\left(x + 3\right)^{2} + \left(y - 2\right)^{2} = 49$,

(iii) $x^{2} + \left(y + 4\right)^{2} = 144$.

On separate diagrams sketch the circles with the following equations

(i) $x^{2}+y^{2}=9$

(ii) $\left(x - 4\right)^{2} + \left(y - 3\right)^{2} = 4^{2}$

(i) Complete the square of $x^{2}+4x$.

(ii) Complete the square of $y^{2}-6y$.

(i) Use your answers to part (a) to show that the equation $x^{2}+y^{2}+4x-6y+4=0$ can be written in the form $\left(x + 2\right)^{2} + \left(y - 3\right)^{2} = 9$.

(ii) Hence, write down the centre and the radius of the circle with equation $x^{2}+y^{2}+4x-6y+4=0.$

The line segment connecting the two points (1 , 0) and (9 , 4) is the diameter of a circle.

Find the centre and radius of the circle.

Determine if the circles with equations

$\left(x + 4\right)^{2} + y^{2} = 9 \text{ } and \left( x - 2\right)^{2} + y^{2} = 9$

intersect once, twice or not at all. Fully explain your answer.

On a sketch show how a circle and a line can either have 0, 1 or 2 intersections.

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

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.

$C_{2}$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 A C B equals 90 degree.

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 $ax^{2}+ay^{2}+bx+cy+d=0$, 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 $C_{2}$.