Equation of a Circle (Edexcel GCSE Maths: Higher): Exam Questions

The equation of circle $\mathbf{C}$ is $x^2+y^2=16$

The circle $\mathbf{C}$ is translated by the vector $\left(\begin{array}{c}3\\ 0\end{array}\right)$ to give circle $\mathbf{B}$.

Draw a sketch of circle $\mathbf{B}$.

Label with coordinates  

• the centre of circle $\mathbf{B}$    

• and any points of intersection with the $x$-axis.

Construct the graph of $x^2+y^2=9$

By drawing the line $x+y=1$ on the grid, solve the equations

 $$\begin{gathered} x^2+y^2=9 \\ x+y=1 \end{gathered}$$

On the grid, construct the graph of $x^2+y^2=16$

Find estimates for the solutions of the simultaneous equations

$\begin{array}{rcl}{x}^2+y^2& =& 16\\ y& =& 2x+1\end{array}$

$A,B$ and $C$ are points on the circle    $x^2+y^2=36$   as shown.

$A$ is on the $y$-axis.
$B$ is on the $x$-axis.
$M$ is the midpoint of $AB$.
$COM$ is a straight line.

Show that the coordinates of $A$ are (0, 6)

Work out the coordinates of $B$.

Show that the equation of the straight line passing through $C,O$and $M$ is $y=x$

Work out the coordinates of $C$. Give your answers in surd form.

$P$(–1, 4) is a point on a circle, centre $O$

Work out the equation of the tangent to the circle at $P$.

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

Here is a sketch of the circle  $x^2+y^2=45$

Show that the tangent to this circle at the point (-3, 6) has a gradient of $\frac{1}{2}$

Find the equation of the tangent at the point (-3, 6).

The point ( -5, 2 ) lies on the circumference of a circle, centre (0, 0).

Find the equation of the circle.

Work out the gradient of the tangent to the circle at (-5, 2).

In this question all units are in cm. A circle has equation $x^2+y^2=36$

Write down the radius and centre of the circle.

radius: .......................................................... cm

centre: (..........................., ...........................)     

The distinct points A ($a,\sqrt{11}$) and B ($b,\sqrt{11}$) lie on the circumference of the circle.

Work out the length AB.

................................................... cm 

On the grid, draw the graph of $x^2+y^2=64$

Use your graph to find estimates for the solutions of the simultaneous equations

$x^2+y^2=64$

$2x=y+8$

Here is a circle, centre $O$, and the tangent to the circle at the point $P$(4, 3) on the circle.

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

On the grid, draw the graph of $x^2+y^2=12.25$

Hence find estimates for the solutions of the simultaneous equation

$\begin{array}{rcl}{x}^2+y^2& =& 12.25\\ 2x+y& =& 1\end{array}$

$\mathbf{L}$ is the circle with equation $x^2+y^2=4$

$P\left(\frac{3}{2},\frac{\sqrt{7}}{2}\right)$ is a point on $\mathbf{L}$.

Find an equation of the tangent to $\mathbf{L}$ at the point $P$.

The diagram shows a circle, centre the origin.

Write down the equation of the circle.

Point $\mathrm{P}$ has coordinates (8, —6).
Show that point $\mathrm{P}$ lies on the circle.

Find the equation of the tangent to the circle at point $\mathrm{P}$.

A circle has equation x² + y² = 80. 

Calculate the diameter of the circle.
Give your answer as a surd in its simplest form.

Colin says that the point (5, 7) lies outside the circle.

Is Colin correct?
Show your reasoning.

Show that the line with equation y = $\frac{1}{2}$ x + 10 is a tangent to the circle.

$P$ is a point on the circle with equation $x^2+y^2=80$

$P$ has $x$-coordinate 4 and is below the $x$-axis.

Work out the equation of the tangent to the circle at $P$.

The diagram shows a circle, centre $O$, with equation $y^2+x^2=169$

A tangent meets the circle at point, $Q$, with coordinates $\left(-5,y\right)$.

Calculate the value of $y$

Calculate the gradient of the tangent to the circle at point $Q$.

The diagram shows the circle    $x^2+y^2=10$

$P$ lies on the circle and has $x$-coordinate 1 The tangent at$P$ intersects the $x$-axis at $Q$.

Work out the coordinates of Q.

The diagram shows a circle with centre (0, 0) and a tangent at the point (–12, 16).

The tangent crosses the y-axis at the point (0, p).

Find the value of p.

p = .....................................................

The diagram shows a circle, centre $O$.

The circumference of the circle is 20π cm. Find the equation of the circle.

The line 10x + py = q is a tangent at the point (5, 4) in another circle with centre (0, 0).

Find the value of p and the value of q.

The line $l$ is a tangent to the circle $x^2+y^2=40$ at the point $A$.

$A$ is the point (2, 6).

The line $l$ crosses the $x$-axis at the point $P$.

Work out the area of triangle $OAP$.

The diagram shows the circle x² + y² =5.

Mandy says that the point (2, 1.5) lies inside the circle.

Is she correct?
Show how you decide.

The tangent to the circle at the point P (-2, 1) intersects the y-axis at A.

Show that the area of the triangle APO is 5 square units.

The diagram shows a circle, centre $O$.

$AB$ is the tangent to the circle at the point $A$. Angle $OBA={30}^{\circ }$

Point $B$ has coordinates (16, 0)
Point $P$ has coordinates (3p, p)

Find the value of p.
Give your answer correct to 1 decimal place.
You must show all your working.

Prove algebraically that the straight line with equation $x-2y=10$ is a tangent to the circle with equation $x^2+y^2=20$

$P$ (4, 8) is a point on a circle, centre $O$.

The tangent at $P$ intersects the axes at points $A$ and $B$.

Show that the gradient of the tangent is $-\frac{1}{2}$

Work out the length $AB$.

Give your answer in the form $a\sqrt{5}$ where $a$ is an integer.

You must show your working.

....................................units

In this question, all lengths are in centimetres.

$A$ is a point on a circle, centre $O$.
$B$ is a point on a different circle, centre $O$.
$AB=20$

The equation of the larger circle is    $x^2+y^2=144$
radius of smaller circle : radius of larger circle   = 4 : 5

Work out the size of angle $AOB$

............................................ degrees