1.03d Circles: equation (x-a)^2+(y-b)^2=r^2

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.

12 Fig. 12 shows the circle \(( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 25\), the line \(4 y = 3 x - 32\) and the tangent to the circle at the point \(\mathrm { A } ( 5,2 )\). D is the point of intersection of the line \(4 y = 3 x - 32\) and the tangent at A . \begin{figure}[h] \end{figure}

1. Write down the coordinates of C , the centre of the circle.
2. (A) Show that the line \(4 y = 3 x - 32\) is a tangent to the circle.
   (B) Find the coordinates of B , the point where the line \(4 y = 3 x - 32\) touches the circle.
3. Prove that ADBC is a square.
4. The point E is the lowest point on the circle. Find the area of the sector ECB .

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 .

7 The parametric equations of a circle are \(x = 7 + 5 \cos \theta , \quad y = 5 \sin \theta - 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).

1. Find a cartesian equation of the circle.
2. State the coordinates of the centre of the circle. Answer all the questions.
   Section B (77 marks)

8 The curves \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) and \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) crosses the \(x\)-axis at D and the \(y\)-axis at C . \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-7_826_819_520_255}

1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
2. Determine the coordinates of A, B, C and D.
3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(\mathrm { y } = \mathrm { mx } + c\), where \(m\) and \(c\) are constants to be determined.
4. Points A , B , C and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\), \(b\) and \(r ^ { 2 }\) are constants to be determined.

3 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 12 x - 6 y + 20 = 0\).

1. By completing the square, express the equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle.
3. The line with equation \(y = x + 4\) intersects the circle at the points \(P\) and \(Q\).
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } - 5 x + 6 = 0$$
   2. Find the coordinates of \(P\) and \(Q\).

5 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x + 6 y = 11\).

1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. The point \(O\) has coordinates \(( 0,0 )\).
   1. Find the length of CO .
   2. Hence determine whether the point \(O\) lies inside or outside the circle, giving a reason for your answer.

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0\).

1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. The point \(D\) has coordinates (7, -2).
   1. Verify that the point \(D\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(D\), giving your answer in the form \(m x + n y = p\), where \(m , n\) and \(p\) are integers.
4. 1. A line has equation \(y = k x\). Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$
   2. Find the values of \(k\) for which the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (d)(ii).

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 4 x + 12 y + 15 = 0\).

1. Find:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. Explain why the circle lies entirely below the \(x\)-axis.
3. The point \(P\) with coordinates \(( 5 , k )\) lies outside the circle.
   1. Show that \(P C ^ { 2 } = k ^ { 2 } + 12 k + 45\).
   2. Hence show that \(k ^ { 2 } + 12 k + 20 > 0\).
   3. Find the possible values of \(k\).

6 A circle has centre \(C ( - 3,1 )\) and radius \(\sqrt { 13 }\).

1. 1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
   2. Hence find the equation of the circle in the form $$x ^ { 2 } + y ^ { 2 } + m x + n y + p = 0$$ where \(m , n\) and \(p\) are integers.
2. The circle cuts the \(y\)-axis at the points \(A\) and \(B\). Find the distance \(A B\).
3. 1. Verify that the point \(D ( - 5 , - 2 )\) lies on the circle.
   2. Find the gradient of \(C D\).
   3. Hence find an equation of the tangent to the circle at the point \(D\).

6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.

1. The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
2. The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
3. Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
4. Hence determine the possible values of the width of the garden. \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
   1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
   2. Write down:
      1. the coordinates of \(C\);
      2. the radius of the circle.
      3. Sketch the circle.
      4. A line has equation \(y = k x + 6\), where \(k\) is a constant.
         1. Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
         2. The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
         3. Hence find the values of \(k\) for which the line is a tangent to the circle.

7 A circle with centre \(C ( - 3,2 )\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 12$$

1. Find the \(y\)-coordinates of the points where the circle crosses the \(y\)-axis.
2. Find the radius of the circle.
3. The point \(P ( 2,5 )\) lies outside the circle.
   1. Find the length of \(C P\), giving your answer in the form \(\sqrt { n }\), where \(n\) is an integer.
   2. The point \(Q\) lies on the circle so that \(P Q\) is a tangent to the circle. Find the length of \(P Q\).

3 A circle has centre \(C ( 2 , - 1 )\) and radius 5 . The point \(P\) has coordinates \(( 6,2 )\).

1. Write down an equation of the circle.
2. Verify that the point \(P\) lies on the circle.
3. Find the gradient of the line \(C P\).
4. 1. Find the gradient of a line which is perpendicular to \(C P\).
   2. Hence find an equation for the tangent to the circle at the point \(P\).