Equation of a circle and tangents – Year 11 GCSE maths worksheet with answers to download as PDF

The equation of a circle with centre \((a,b)\) and radius \(r\) is \((x-a)^2+(y-b)^2=r^2\). Write the equation of each circle.

1. Centre \((0,0)\), radius 5. [1 mark]
2. Centre \((2,3)\), radius 4. [2 marks]
3. Centre \((-1,4)\), radius 3. [2 marks]
4. Centre \((0,0)\), radius \(\sqrt{20}\). [1 mark]
5. Centre \((3,-5)\), passing through the point \((6,-1)\). [3 marks]

Write down the centre and the radius of each circle. Give radii as surds in simplest form when they are not whole numbers.

1. \(x^2+y^2=49\) [2 marks]
2. \((x-3)^2+(y+2)^2=25\) [2 marks]
3. \((x+4)^2+y^2=12\) [2 marks]
4. \((x-1)^2+(y-1)^2=18\) [2 marks]

A circle has equation \(x^2+y^2=25\).

1. Work out \(x^2+y^2\) for each point. [3 marks]
2. State whether each point lies on, inside or outside the circle. [2 marks]
3. Which point is closest to the centre? [1 mark]

The circle \(x^2+y^2=25\) passes through \(A(3,4)\). The tangent at \(A\) is perpendicular to the radius \(OA\).

1. Find the gradient of the radius \(OA\). [1 mark]
2. Write down the gradient of the tangent at \(A\). [1 mark]
3. Find the equation of the tangent in the form \(ax+by=c\). [3 marks]

The point \(P(-6,8)\) lies on the circle \(x^2+y^2=100\).

1. Show that \(P\) is on the circle. [1 mark]
2. Find the gradient of the tangent at \(P\). [2 marks]
3. Find the equation of the tangent in the form \(y=mx+c\). [3 marks]

The circle \(x^2+y^2=25\) crosses the axes at four points.

1. Write down the four points. [2 marks]
2. Write the equation of the tangent at \((5,0)\) and explain why it is vertical. [2 marks]
3. Write the equation of the tangent at \((0,-5)\). [2 marks]
4. The four tangents form a shape. What is it and what is its area? [2 marks]

A big wheel has a radius of 10 m and its centre is 10 m above the ground. Take the ground as the line \(y=0\) and the centre as \(C(0,10)\).

1. Write the equation of the circle traced out by the passenger pods. [2 marks]
2. A pod is 6 m to the right of the centre line. Find the two possible heights. [4 marks]
3. Show that the pod at \((6,2)\) is on the wheel. [2 marks]
4. What is the lowest point of the wheel and how far is it from the ground? [2 marks]

A radar at the origin \(O\) detects ships within a distance of 50 km. All distances are in kilometres.

1. Write the equation of the boundary of the radar range. [1 mark]
2. Is a ship at \((30,40)\) on the boundary, inside or outside? [2 marks]
3. Is a ship at \((35,40)\) detected? How far is it beyond the edge of the range, to 2 decimal places? [3 marks]
4. Is a ship at \((20,-30)\) detected? [2 marks]

A sprinkler at the point \((3,-2)\) waters a circular area of radius 6 m. The lawn is on a coordinate grid in metres.

1. Write the equation of the boundary of the watered area. [2 marks]
2. Show that \((3,4)\) is on the boundary. [2 marks]
3. Find the other boundary point on the line \(y=-2\) on the right of the sprinkler. [2 marks]
4. A flowerbed is at \((6,2)\) and another at \((8,3)\). Which one is watered? [3 marks]

A circular running track has centre \(O\) at the origin and radius 40 m. A runner at \(A(24,32)\) loses control and runs along the tangent.

1. Show that \(A\) is on the track. [2 marks]
2. Find the equation of the tangent at \(A\) in the form \(3x+4y=c\). [4 marks]
3. Where does the runner cross the \(x\)-axis and the \(y\)-axis? [3 marks]
4. How far does the runner travel from \(A\) to the \(y\)-axis? Give your answer to 1 decimal place. [2 marks]

A straight path with equation \(y=x+1\) crosses a circular pond with equation \(x^2+y^2=25\).

1. Show that the \(x\)-coordinates of the points where the path meets the pond satisfy \(x^2+x-12=0\). [3 marks]
2. Solve the equation and find the two points. [3 marks]
3. Work out the length of the path inside the pond to 2 decimal places. [3 marks]

The circle \(x^2+y^2=8\) and the line \(y=x+4\) are drawn on the same axes.

1. Show that the line meets the circle where \(x^2+4x+4=0\). [3 marks]
2. Solve this equation and explain why this shows that the line is a tangent. [3 marks]
3. Find the point of contact. [1 mark]
4. Check that the radius to the point of contact is perpendicular to the line. [3 marks]

A circular lake has equation \(x^2+y^2=25\) (units in metres). A hiker stands at \(T(13,0)\) and walks along a straight line to a point \(A\) on the lake so that \(TA\) is a tangent.

1. Explain why triangle \(OAT\) has a right angle at \(A\). [1 mark]
2. Work out the length \(TA\). [3 marks]
3. Work out angle \(ATO\) to 1 decimal place. [2 marks]
4. Work out the area of kite \(OATB\) where \(TB\) is the other tangent. [2 marks]

A circular pond has three markers on its edge at \(O(0,0)\), \(A(8,0)\) and \(B(0,6)\), with distances in metres.

1. Explain why \(AB\) is a diameter of the pond. [2 marks]
2. Find the centre and the radius of the pond. [3 marks]
3. Write the equation of the edge of the pond. [2 marks]
4. Show that the point \((4,8)\) is on the edge and work out the area of the pond to 1 decimal place. [3 marks]

A circle has equation \((x-1)^2+(y-2)^2=20\). The point \(P(5,4)\) lies on the circle.

1. Show that \(P\) is on the circle. [1 mark]
2. Find the gradient of the radius from the centre \(C(1,2)\) to \(P\). [2 marks]
3. Find the equation of the tangent at \(P\). [3 marks]
4. The tangent meets the axes at \(A\) and \(B\). Work out the area of triangle \(OAB\). [4 marks]

The circle \(x^2+y^2=40\) meets the line \(y=-x+8\) at \(A\) and \(B\).

1. Show that the points \(A(6,2)\) and \(B(2,6)\) are on both the line and the circle. [4 marks]
2. Work out the length of the chord \(AB\) to 2 decimal places. [2 marks]
3. Find the midpoint \(M\) of \(AB\) and show that \(OM\) is perpendicular to \(AB\). [3 marks]
4. Work out the distance \(OM\) and verify with Pythagoras. [3 marks]

\(A(-2,3)\) and \(B(6,9)\) are the ends of a diameter of a circle.

1. Find the centre of the circle. [2 marks]
2. Work out the radius. [3 marks]
3. Write the equation of the circle. [2 marks]
4. Show that the point \((2,11)\) lies on the circle. [2 marks]

The circle \(x^2+y^2=25\) has two tangents from the point \(P(0,10)\), touching the circle at \(A\) and \(B\).

1. Work out the length of each tangent. [3 marks]
2. Work out angle \(OPA\). [2 marks]
3. Explain why angle \(APB=60^\circ\). [2 marks]
4. Work out the area of the quadrilateral \(OAPB\) to 2 decimal places. [3 marks]

The line \(y=2x+k\) is a tangent to the circle \(x^2+y^2=5\).

1. Show that the line meets the circle where \(5x^2+4kx+k^2-5=0\). [3 marks]
2. For a tangent the equation has exactly one solution, so the discriminant is zero. Show that \(k^2=25\). [4 marks]
3. Find the two possible values of \(k\) and the points of contact when \(k=5\). [3 marks]

A circle has equation \(x^2+y^2-6x-8y=0\).

1. Complete the square to show that the centre is \((3,4)\) and the radius is 5. [4 marks]
2. Show that the origin is on the circle and find the equation of the tangent there. [4 marks]
3. The circle crosses the axes again at \(A\) and \(B\). Find them and show that \(AB\) is a diameter. [4 marks]