1.03e Complete the square: find centre and radius of circle

10 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 8 = 0\).

1. Find the centre and radius of the circle.
2. The circle passes through the point \(( - 3 , k )\), where \(k < 0\). Find the value of \(k\).
3. Find the coordinates of the points where the circle meets the line with equation \(x + y = 6\).

2

1. Write down the equation of the circle with centre \(( 0,0 )\) and radius 7 .
2. A circle with centre \(( 3,5 )\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 10 y - 30 = 0\). Find the radius of the circle.

9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .

1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
2. Calculate the length of \(A B\), giving your answer in simplified surd form.
3. Find an equation for the line \(A B\).

9

1. Find the equation of the circle with radius 10 and centre ( 2,1 ), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. The circle passes through the point \(( 5 , k )\) where \(k > 0\). Find the value of \(k\) in the form \(p + \sqrt { q }\).
3. Determine, showing all working, whether the point \(( - 3,9 )\) lies inside or outside the circle.
4. Find an equation of the tangent to the circle at the point ( 8,9 ).

12 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).

1. Show that the centre of this circle is \(\mathrm { C } ( 4,2 )\) and find the radius of the circle.
2. Show that the origin lies inside the circle.
3. Show that AB is a diameter of the circle, where A has coordinates (2, 7) and B has coordinates \(( 6 , - 3 )\).
4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).

11 \begin{figure}[h] \end{figure} A circle has centre \(C ( 1,3 )\) and passes through the point \(A ( 3,7 )\) as shown in Fig. 11.

1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.

12

1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.

10 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.

6 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 2 x - 8 = 0\).
Find the centre and radius of the circle.

12 You are given that the equation of the circle shown in Fig. 12 is $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$$ \begin{figure}[h] \end{figure}

1. Show that the centre, Q , of the circle is \(( 2,3 )\) and find the radius.
2. The circle crosses the \(x\)-axis at B and C . Show that the coordinates of C are \(( 6,0 )\) and find the coordinates of B .
3. Find the gradient of the line QC and hence find the equation of the tangent to the circle at C.
4. Given that M is the mid-point of BC , find the coordinates of the point where QM meets the tangent at C .

13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h] \end{figure}

1. Find the coordinates of X .
2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
   Write down the equation of the circumcircle of the triangle ABC .
4. Find the coordinates of the points where the circle cuts the \(x\) axis.

3. A circle has the equation $$x ^ { 2 } + y ^ { 2 } - 6 y - 7 = 0$$

1. Find the coordinates of the centre of the circle.
2. Find the radius of the circle.

9. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).

1. Find an equation for \(C\).
2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).

1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
2. Find the \(x\)-coordinates of \(A\) and \(B\).
3. Show that \(A B = 2 \sqrt { 10 }\).

10. \includegraphics[max width=\textwidth, alt={}, center]{af6fdbed-fcab-4db8-9cdf-fd049ce720fd-3_668_787_918_431} The diagram shows the circle \(C\) and the straight line \(l\).
The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).

1. Find the gradient of 1 .
2. Find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the centre of \(C\).
4. Show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0$$

7. A circle has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).

1. Find the length of the diameter of the circle.
2. Find an equation for the circle.
3. Show that the line \(y = 2 x - 3\) is a tangent to the circle and find the coordinates of the point of contact.

8. The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0\) and the straight line with equation \(y = 2 x - 3\).

1. Find the coordinates of the centre and the radius of the circle.
2. Find the coordinates of the point on the line which is closest to the circle.

9. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\).
3. Sketch \(C\). Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).

2 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h] \end{figure}

1. Write down the coordinates of B .
2. Find the radius of the circle and hence write down the equation of the circle.
3. AD is a diameter of the circle. Find the coordinates of D .
4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\).

3 The circle \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\) has centre C.

1. Write down the radius of the circle and the coordinates of C .
2. Find the coordinates of the intersections of the circle with the \(x\) - and \(y\)-axes.
3. Show that the points \(\mathrm { A } ( 1,6 )\) and \(\mathrm { B } ( 7,4 )\) lie on the circle. Find the coordinates of the midpoint of AB . Find also the distance of the chord AB from the centre of the circle.

5 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0$$
4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle.

4 A circle has equation \(( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\).

1. State the coordinates of the centre and the radius of this circle.
2. State, with a reason, whether or not this circle intersects the \(y\)-axis.
3. Find the equation of the line parallel to the line \(y = 2 x\) that passes through the centre of the circle.
4. Show that the line \(y = 2 x + 2\) is a tangent to the circle. State the coordinates of the point of contact.

2 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).

1. Show that the centre of this circle is \(C ( 4,2 )\) and find the radius of the circle.
2. Show that the origin lies inside the circle.
3. Show that AB is a diameter of the circle, where A has coordinates ( 2,7 ) and B has coordinates \(( 6 , - 3 )\).
4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).

3 \begin{figure}[h] \end{figure} Not to scale A circle has centre \(\mathrm { C } ( 1,3 )\) and passes through the point \(\mathrm { A } ( 3,7 )\) as shown in Fig. 11.

1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.

6 The points \(\mathrm { A } ( - 1,6 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 13,4 )\) are joined by straight lines.

1. Prove that the lines AB and BC are perpendicular.
2. Find the area of triangle ABC .
3. A circle passes through the points A , B and C . Justify the statement that AC is a diameter of this circle. Find the equation of this circle.
4. Find the coordinates of the point on this circle that is furthest from B.