Equation of a Circle (Edexcel GCSE Maths: Higher)

The equation of a curve is 
is the point where the curve intersects the -axis.

State the coordinates of .

The equation of circle  is 

The circle is translated by the vector  to give circle .

Draw a sketch of circle .

Label with coordinates  
the centre of circle    
and any points of intersection with the -axis.

Construct the graph of 

By drawing the line  on the grid, solve the equations 

On the grid, construct the graph of 

Find estimates for the solutions of the simultaneous equations

and  are points on the circle       as shown.

is on the -axis.
is on the -axis.
is the midpoint of .
 is a straight line.

Show that the coordinates of are (0, 6)

Work out the coordinates of .

Show that the equation of the straight line passing through and  is 

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

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

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

Give your answer in the form   

Here is a sketch of the circle 

Show that the tangent to this circle at the point (-3, 6) has a gradient of 

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

Write down the radius and centre of the circle.

radius: .......................................................... cm centre: (..........................., ...........................)     

The distinct points A () and B () lie on the circumference of the circle.

Work out the length AB.

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

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

Find an equation of the tangent at the point .

On the grid, draw the graph of 

Hence find estimates for the solutions of the simultaneous equation

is the circle with equation 

 is a point on .

Find an equation of the tangent to at the point .

The diagram shows a circle, centre the origin.

Write down the equation of the circle.

Point has coordinates (8, —6).
Show that point lies on the circle.

Find the equation of the tangent to the circle at point .

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 = x + 10 is a tangent to the circle.

is a point on the circle with equation 

has -coordinate 4 and is below the -axis.

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

The diagram shows the circle   

lies on the circle and has -coordinate 1 The tangent at intersects the -axis at .

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 .

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 is a tangent to the circle  at the point . is the point (2, 6).

The line crosses the -axis at the point .

Work out the area of triangle .

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 .

is the tangent to the circle at the point . Angle 

Point has coordinates (16, 0)
Point 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  is a tangent to the circle with equation 

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

The tangent at intersects the axes at points and .

Show that the gradient of the tangent is 

Work out the length .

Give your answer in the form where is an integer.

You must show your working.

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

In this question, all lengths are in centimetres.

is a point on a circle, centre .
is a point on a different circle, centre .

The equation of the larger circle is    
radius of smaller circle : radius of larger circle   = 4 : 5

Work out the size of angle

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