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

11. \begin{figure}[h] \end{figure} Figure 3 shows the circle \(C\) with equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 32 = 0$$ and the line \(l\) with equation $$2 y + x + 6 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\).
2. Find the shortest distance between \(C\) and \(l\).

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\). Given that \(C\) does not cut or touch the \(x\)-axis,
2. find the range of possible values for \(k\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the circle \(C\)

• the point \(P ( - 1 , k + 8 )\) is the centre of \(C\)
• the point \(Q \left( 3 , k ^ { 2 } - 2 k \right)\) lies on \(C\)
• \(k\) is a positive constant
• the line \(l\) is the tangent to \(C\) at \(Q\)

Given that the gradient of \(l\) is - 2

1. show that $$k ^ { 2 } - 3 k - 10 = 0$$
2. Hence find an equation for \(C\)

A circle \(C\) has centre \(( 2,5 )\). Given that the point \(P ( - 2,3 )\) lies on \(C\).

1. find an equation for \(C\). The line \(l\) is the tangent to \(C\) at the point \(P\). The point \(Q ( 2 , k )\) lies on \(l\).
2. Find the value of \(k\).

3. A circle \(C\) has equation $$x ^ { 2 } - 22 x + y ^ { 2 } + 10 y + 46 = 0$$ a. Find
i. the coordinates of the centre \(A\) of the circle
ii. the radius of the circle. Given that the points \(Q ( 5,3 )\) and \(S\) lie on \(C\) such that the distance \(Q S\) is greatest,
b. find an equation of tangent to \(C\) at \(S\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are constants to be found.

1. A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 14 y = 40\).

The line \(l\) has equation \(y = x + k\), where \(k\) is a constant.
a. Show that the \(x\)-coordinate of the points where \(C\) and \(l\) intersect are given by the solutions to the equation $$2 x ^ { 2 } + ( 2 k - 20 ) x + k ^ { 2 } - 14 k - 40 = 0$$ b. Hence find the two values of \(k\) for which \(l\) is a tangent to \(C\).

1. A circle has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 16 y = 80$$

1. Find
   1. the coordinates of the centre of the circle,
   2. the radius of the circle. Given that \(P\) is the point on the circle that is furthest away from the origin \(O\),
2. find the exact length \(O P\)

1. A circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 6 k x - 2 k y + 7 = 0$$ where \(k\) is a constant.

1. Find in terms of \(k\),
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\) The line with equation \(y = 2 x - 1\) intersects \(C\) at 2 distinct points.
2. Find the range of possible values of \(k\).

6. Figure 1 Figure 1 shows a sketch of triangle \(A B C\).
Given that

• \(\overrightarrow { A B } = - 3 \mathbf { i } - 4 \mathbf { j } - 5 \mathbf { k }\)
• \(\overrightarrow { B C } = \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
  1. find \(\overrightarrow { A C }\)
  2. show that \(\cos A B C = \frac { 9 } { 10 }\)

2. \begin{figure}[h] \end{figure} The shape \(A B C D O A\), as shown in Figure 1, consists of a sector \(C O D\) of a circle centre \(O\) joined to a sector \(A O B\) of a different circle, also centre \(O\). Given that arc length \(C D = 3 \mathrm {~cm} , \angle C O D = 0.4\) radians and \(A O D\) is a straight line of length 12 cm ,

1. find the length of \(O D\),
2. find the area of the shaded sector \(A O B\).

1. The circle \(C _ { 1 }\) has equation

$$x ^ { 2 } + y ^ { 2 } - 6 x + 14 y + 33 = 0$$

1. Find
   1. the coordinates of the centre of \(C _ { 1 }\)
   2. the radius of \(C _ { 1 }\) A different circle \(C _ { 2 }\)
      Given that \(C _ { 1 }\) and \(C _ { 2 }\) intersect at 2 distinct points,
2. find the range of values of \(k\), writing your answer in set notation.

1. A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2 x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
2. find the two possible values of \(r\), giving your answers as fully simplified surds.

A circle with centre \(A ( 3 , - 1 )\) passes through the point \(P ( - 9,8 )\) and the point \(Q ( 15 , - 10 )\)

1. Show that \(P Q\) is a diameter of the circle.
2. Find an equation for the circle. A point \(R\) also lies on the circle. Given that the length of the chord \(P R\) is 20 units,
3. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
4. Find the size of angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.

2 The circle \(x ^ { 2 } + y ^ { 2 } - 4 x + k y + 12 = 0\) has radius 1.
Find the two possible values of the constant \(k\).

7

1. In this question you must show detailed reasoning. Find the range of values of the constant \(m\) for which the simultaneous equations \(y = m x\) and \(x ^ { 2 } + y ^ { 2 } - 6 x - 2 y + 5 = 0\) have real solutions.
2. Give a geometrical interpretation of the solution in the case where \(m = 2\).

7 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-05_848_1049_260_242} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 9 y + 19 = 0\) and centre \(C\).

1. Find the following.
   The tangent to the circle at \(D\) meets the \(x\)-axis at the point \(A \left( \frac { 55 } { 4 } , 0 \right)\) and the \(y\)-axis at the point \(B ( 0 , - 11 )\).
2. Determine the area of triangle \(O B D\).

4 The circle \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + k = 0\) has radius 5.
Determine the value of \(k\).

6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).

1. Show that \(A B = B C\).
2. Show that angle \(A B C\) is not a right angle.
3. Find the coordinates of the midpoint of \(A C\).
4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
6. Find the coordinates of this second point.

8 In this question you must show detailed reasoning.

1. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 4 y - 20 = 0\).
2. Find the points of intersection of the circle with the line \(x + 3 y - 10 = 0\).

8 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 16 y + 48 = 0\).

1. Find the coordinates of C . A line has equation \(\mathrm { y } = \mathrm { x } - 2\) and intersects the circle at the points A and B . The midpoints of AC and BC are \(\mathrm { A } ^ { \prime }\) and \(\mathrm { B } ^ { \prime }\) respectively.
2. Determine the exact distance \(\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }\).

8 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 25\).

1. Write down

4 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 8 x - 6 y - 39 = 0\).

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

15 The circle \(x ^ { 2 } + y ^ { 2 } + 2 x - 14 y + 25 = 0\) has its centre at the point \(C\). The line \(7 y = x + 25\) intersects the circle at points A and B . Prove that triangle ABC is a right-angled triangle.

7 In this question you must show detailed reasoning.
The points \(\mathrm { A } ( - 1,4 )\) and \(\mathrm { B } ( 7 , - 2 )\) are at opposite ends of a diameter of a circle.

1. Find the equation of the circle.
2. Find the coordinates of the points of intersection of the circle and the line \(y = 2 x + 5\).
3. Q is the point of intersection with the larger \(y\)-coordinate. Calculate the area of the triangle ABQ .