Straight lines and circles – Year 12 A level maths worksheet with answers to download as PDF

The points \(A\) and \(B\) have coordinates \(A(2,-1)\) and \(B(8,7)\).

1. Find the gradient of \(AB\). [1 mark]
2. Find the equation of the line \(AB\) in the form \(ax+by+c=0\) where \(a\), \(b\), \(c\) are integers. [3 marks]
3. Find the midpoint \(M\) of \(AB\). [1 mark]
4. Find the length \(AB\). [2 marks]

The line \(L\) has equation \(3x+4y=12\).

1. Find the gradient of \(L\). [1 mark]
2. Find the equation of the line parallel to \(L\) through \((4,-2)\). [2 marks]
3. Find the equation of the line perpendicular to \(L\) through \((0,5)\). [2 marks]
4. Find the coordinates of the foot of the perpendicular from \((0,5)\) to \(L\) and the distance from \((0,5)\) to \(L\). [4 marks]

A circle has centre \(C(3,-2)\) and radius 5.

1. Write down its equation in the form \((x-a)^2+(y-b)^2=r^2\). [2 marks]
2. Expand to the form \(x^2+y^2+px+qy+s=0\). [2 marks]
3. Show that the points \((6,2)\) and \((0,2)\) lie on the circle. [2 marks]
4. Determine whether \((7,1)\) lies inside, outside or on the circle. [2 marks]

Find the centre and radius of each circle.

1. \(x^2+y^2-8x+6y-11=0\) [3 marks]
2. \(x^2+y^2+2x-4y-4=0\) [3 marks]
3. Find the distance between the two centres and decide whether the circles intersect. [3 marks]

\(A(1,2)\) and \(B(7,10)\) are the ends of a diameter of a circle.

1. Find the centre and the radius. [3 marks]
2. Write down the equation of the circle. [1 mark]
3. Show that \(C(8,9)\) lies on the circle. [2 marks]
4. Use gradients to show that angle \(ACB\) is a right angle and name the circle theorem. [3 marks]

The circle \(C\) has equation \((x-1)^2+(y+2)^2=25\).

1. Show that \(P(4,2)\) lies on \(C\). [1 mark]
2. Find the gradient of the radius to \(P\). [1 mark]
3. Find the equation of the tangent at \(P\) in the form \(ax+by=c\). [3 marks]
4. The tangent meets the \(x\)-axis at \(T\). Find the coordinates of \(T\) and the area of the triangle formed by the tangent and the two axes. [4 marks]

On a map (units in km) a phone mast is at \(M(3,4)\). It gives coverage within 5 km. A straight road is the line \(y=x-2\).

1. Write down the equation of the boundary of the coverage area. [1 mark]
2. Show that the road enters the coverage area where \(x^2-9x+10=0\). [4 marks]
3. Find the length of road within the coverage area, to 2 decimal places. [3 marks]
4. Find the shortest distance from the mast to the road. [3 marks]

A garden plan uses metres. A circular pond is the circle \((x-6)^2+(y-5)^2=9\). A straight path is the line \(3x+4y=53\).

1. Write down the centre and radius of the pond. [2 marks]
2. Show that the path is a tangent to the pond. [5 marks]
3. Find the coordinates of the point where the path touches the pond. [3 marks]
4. Find the equation of the other tangent to the pond of the form \(3x+4y=k\). [2 marks]

A Ferris wheel has its centre 12 m above the ground and a radius of 10 m. Take the origin at ground level directly below the centre, with \(x\) horizontal and \(y\) the height above the ground.

1. Write down the equation of the circle that the cabins follow. [2 marks]
2. A cabin is 6 m horizontally from the centre. Find its two possible heights. [3 marks]
3. What is the lowest height of a cabin? [1 mark]
4. A cabin is 5 m horizontally from the centre and in the upper half. Find its height to 1 decimal place. [2 marks]

Three lamp posts stand at \(A(0,0)\), \(B(8,0)\) and \(C(0,6)\) on a plan (metres). A circular fountain must pass through all three posts.

1. Find the equations of the perpendicular bisectors of \(AB\) and \(AC\). [3 marks]
2. Hence find the centre of the circle and its radius. [3 marks]
3. Write down the equation of the circle in the form \(x^2+y^2+px+qy+s=0\). [3 marks]
4. Which side of the triangle is a diameter? Explain. [2 marks]

On a plan a road passes through \((0,2)\) and \((6,5)\). A farmer wants a fence parallel to the road passing through \(F(2,6)\).

1. Find the equation of the road and of the fence. [4 marks]
2. The farmer builds a straight track from \(F\) meeting the road at right angles. Find the equation of the track. [2 marks]
3. Find where the track meets the road. [3 marks]
4. Find the shortest distance from the fence to the road, to 2 decimal places. [2 marks]

Router \(R_1\) covers a circle \((x-2)^2+(y-3)^2=16\) and router \(R_2\) covers the circle \((x-8)^2+(y-3)^2=9\) (units: metres).

1. Find the distance between the routers and show that the coverage areas overlap. [3 marks]
2. Show that the \(x\)-coordinate of the intersection points of the boundaries is \(\tfrac{67}{12}\). [3 marks]
3. Find the two \(y\)-coordinates to 2 decimal places. [3 marks]
4. Find the length of the common chord to 2 decimal places. [1 mark]

A circular lake has equation \((x-2)^2+(y-1)^2=13\) on a map (km). A straight track is built to touch the lake at \(P(5,3)\).

1. Show that \(P\) lies on the lake boundary. [1 mark]
2. Find the equation of the tangent at \(P\). [4 marks]
3. The track runs from the \(x\)-axis to the \(y\)-axis. Find the coordinates of its ends. [2 marks]
4. Find the length of the track, to 2 decimal places. [2 marks]

Two villages are at \(A(1,2)\) and \(B(9,6)\). A new bus stop must be the same distance from each village and must also be on the straight road \(y=x+2\).

1. Find the midpoint and the gradient of \(AB\). [2 marks]
2. Find the equation of the perpendicular bisector of \(AB\). [3 marks]
3. Find the coordinates of the bus stop. [2 marks]
4. Show that the bus stop is 5 km from each village. [2 marks]

A landscape architect wants a circular flower bed passing through the three points \(P(2,1)\), \(Q(6,1)\) and \(R(8,5)\) on a plan (metres).

1. Find the perpendicular bisector of \(PQ\). [2 marks]
2. Find the perpendicular bisector of \(PR\). [3 marks]
3. Find the centre and the exact radius of the circle. [4 marks]
4. Show that \(R\) lies on the circle and find the area of the flower bed to 1 decimal place. [3 marks]

The circle \(C\) has equation \(x^2+y^2=20\). The line \(L\) has equation \(y=2x+k\).

1. Show that the \(x\)-coordinates of the intersections satisfy \(5x^2+4kx+k^2-20=0\). [3 marks]
2. Find the values of \(k\) for which \(L\) is a tangent to \(C\). [3 marks]
3. State the range of \(k\) for which \(L\) crosses \(C\) at two points. [2 marks]
4. For \(k=5\) find the points of intersection and the length of the chord. [4 marks]

The circle \(C\) has equation \(x^2+y^2-6x-4y-12=0\).

1. Find the centre and radius of \(C\). [3 marks]
2. Show that \(P(6,6)\) lies on \(C\) and find the equation of the tangent at \(P\). [4 marks]
3. The tangent meets the \(x\)-axis at \(T\). Find the coordinates of \(T\). [2 marks]
4. Show that \(TP=10\). Then show that the length of the other tangent from \(T\) to \(C\) is also 10. [3 marks]

Two circular lakes on a map (km) have equations \(x^2+y^2=25\) and \((x-6)^2+y^2=25\).

1. State the centres and radii. [2 marks]
2. Show that the lakes meet where \(x=3\) and find the two points. [4 marks]
3. Find the length of the common chord and show that the chord is perpendicular to the line joining the centres. [3 marks]
4. Show that the common chord passes through the midpoint of the line joining the centres. [2 marks]

The circle \(x^2+y^2=25\) has a diameter \(AB\) where \(A(-5,0)\) and \(B(5,0)\).

1. Show that \(P\), \(Q\) and \(R\) lie on the circle. [3 marks]
2. For each point work out the gradients of the lines to \(A\) and to \(B\) and show that they multiply to \(-1\). [6 marks]
3. Let \(T(x,y)\) be any point on the circle other than \(A\) or \(B\). Prove that the gradients of \(TA\) and \(TB\) multiply to \(-1\). [4 marks]

A circle has its centre on the line \(y=x\) and touches the \(x\)-axis. It passes through the point \((8,4)\).

1. Let the centre be \((a,a)\) with \(a>0\). Explain why the radius is \(a\) and write down the equation of the circle. [2 marks]
2. Show that \(a^2-24a+80=0\) and find the two possible circles. [5 marks]
3. Show that the two circles meet at two points and find the second point of intersection. [5 marks]