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

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the circle \(C _ { 1 }\) The points \(A ( 1,4 )\) and \(B ( 7,8 )\) lie on \(C _ { 1 }\) Given that \(A B\) is a diameter of the circle \(C _ { 1 }\)

1. find the coordinates for the centre of \(C _ { 1 }\)
2. find the exact radius of \(C _ { 1 }\) simplifying your answer. Two distinct circles \(C _ { 2 }\) and \(C _ { 3 }\) each have centre \(( 0,0 )\).
   Given that each of these circles touch circle \(C _ { 1 }\)
3. find the equation of circle \(C _ { 2 }\) and the equation of circle \(C _ { 3 }\)

13. The point \(A ( 9 , - 13 )\) lies on a circle \(C\) with centre the origin and radius \(r\).

1. Find the exact value of \(r\).
2. Find an equation of the circle \(C\). A straight line through point \(A\) has equation \(2 y + 3 x = k\), where \(k\) is a constant.
3. Find the value of \(k\). This straight line cuts the circle again at the point \(B\).
4. Find the exact coordinates of point \(B\).

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x + p y + 123 = 0$$ where \(p\) is a constant. Given that the point \(( 1,16 )\) lies on \(C\),

1. find
   1. the value of \(p\),
   2. the coordinates of the centre of \(C\),
   3. the radius of \(C\).
2. Find an equation of the tangent to \(C\) at the point ( 1,16 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-31_33_19_2668_1896}

13. The circle \(C\) has centre \(A ( 1 , - 3 )\) and passes through the point \(P ( 8 , - 2 )\).

1. Find an equation for the circle \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(P\).
2. Find an equation for \(l _ { 1 }\), giving your answer in the form \(y = m x + c\) The line \(l _ { 2 }\), with equation \(y = x + 6\), is the tangent to \(C\) at the point \(Q\).
3. Find the coordinates of the point \(Q\).

14. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 16 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\). Given that the radius of \(C\) is 10
2. find the value of \(k\). The point \(A ( a , - 16 )\), where \(a > 0\), lies on the circle \(C\). The tangent to \(C\) at the point \(A\) crosses the \(x\)-axis at the point \(D\) and crosses the \(y\)-axis at the point \(E\).
3. Find the exact area of triangle \(O D E\).

12. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\)

1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
2. Find an equation for \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
   Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
3. find the length of \(P Q\), giving your answer in its simplest surd form.

6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 14 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The line with equation \(y = k\), where \(k\) is a constant, is a tangent to \(C\).
2. Find the possible values of \(k\). The line with equation \(y = p\), where \(p\) is a negative constant, is a chord of \(C\).
   Given that the length of this chord is 4 units,
3. find the value of \(p\).
   

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9. A circle \(C\) has equation $$( x - k ) ^ { 2 } + ( y - 2 k ) ^ { 2 } = k + 7$$ where \(k\) is a positive constant.

1. Write down, in terms of \(k\),
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\). Given that the point \(P ( 2,3 )\) lies on \(C\)
2. 1. show that \(5 k ^ { 2 } - 17 k + 6 = 0\)
   2. hence find the possible values of \(k\). The tangent to the circle at \(P\) intersects the \(x\)-axis at point \(T\).
      Given that \(k < 2\)
3. calculate the exact area of triangle \(O P T\).

6. \begin{figure}[h] \end{figure} The points \(P ( 23,14 ) , Q ( 15 , - 30 )\) and \(R ( - 7 , - 26 )\) lie on the circle \(C\), as shown in Figure 1.

1. Show that angle \(P Q R = 90 ^ { \circ }\)
2. Hence, or otherwise, find
   1. the centre of \(C\),
   2. the radius of \(C\). Given that the point \(S\) lies on \(C\) such that the distance \(Q S\) is greatest,
3. find an equation of the tangent to \(C\) at \(S\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The point \(P\) lies on \(C\).
      Given that the tangent to \(C\) at \(P\) has equation \(x + 2 y + 10 = 0\)
2. find the coordinates of \(P\)
3. Find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.

2. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 10 y - 21 = 0$$ Find

1. 1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). The point \(P ( 5,4 )\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

1. A circle has equation

$$x ^ { 2 } - 6 x + y ^ { 2 } + 8 y + k = 0$$ where \(k\) is a positive constant. Given that the \(x\)-axis is a tangent to this circle,

1. find the value of \(k\). The circle meets the coordinate axes at the points \(R , S\) and \(T\).
2. Find the exact area of the triangle \(R S T\). \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-21_2647_1840_118_111}

1. A circle \(C\) has centre \(( 2,5 )\)

Given that the point \(P ( 8 , - 3 )\) lies on \(C\)

1. 1. find the radius of \(C\)
   2. find an equation for \(C\)
2. Find the equation of the tangent to \(C\) at \(P\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.

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

$$x ^ { 2 } + y ^ { 2 } + 8 x - 10 y = 29$$

1. 1. Find the coordinates of the centre of \(C _ { 1 }\)
   2. Find the exact value of the radius of \(C _ { 1 }\) In part (b) you must show detailed reasoning.
      The circle \(C _ { 2 }\) has equation $$( x - 5 ) ^ { 2 } + ( y + 8 ) ^ { 2 } = 52$$
2. Prove that the circles \(C _ { 1 }\) and \(C _ { 2 }\) neither touch nor intersect.

4. The points \(P\) and \(Q\) have coordinates \(( - 11,6 )\) and \(( - 3,12 )\) respectively. Given that \(P Q\) is a diameter of the circle \(C\),

1. 1. find the coordinates of the centre of \(C\),
   2. find the radius of \(C\).
2. Hence find an equation of \(C\).
3. Find an equation of the tangent to \(C\) at the point \(Q\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{0e107b51-2fb3-4ad7-8542-5aa0da13b127-13_2255_50_314_34}
   

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6. (i) The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 12 y = k \quad \text { where } k \text { is a constant }$$

1. Find the coordinates of the centre of \(C _ { 1 }\)
2. State the possible range in values for \(k\).
   (ii) The point \(P ( p , 0 )\), the point \(Q ( - 2,10 )\) and the point \(R ( 8 , - 14 )\) lie on a different circle, \(C _ { 2 }\) Given that

7. \begin{figure}[h] \end{figure} The circle with equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$ had centre \(C\) and radius \(r\).

1. Find the coordinates of \(C\).
2. Show that \(r = 5\) The line with equation \(x = 13\) crosses the circle at the points \(P\) and \(Q\) as shown in Figure 1 .
3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). A tangent to the circle from \(O\) touches the circle at point \(X\).
4. Find, in surd form, the length \(O X\). \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-22_2673_1948_107_118}

3. \begin{figure}[h] \end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find

1. the exact length of \(A B\),
2. the coordinates of the midpoint \(P\) of \(A B\),
3. an equation for the circle \(C\).

3. \begin{figure}[h] \end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find

1. the exact length of \(A B\),
2. the coordinates of the midpoint \(P\) of \(A B\),
3. an equation for the circle \(C\).

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the circle \(C\) with centre \(N\) and equation $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = \frac { 169 } { 4 }$$

1. Write down the coordinates of \(N\).
2. Find the radius of \(C\). The chord \(A B\) of \(C\) is parallel to the \(x\)-axis, lies below the \(x\)-axis and is of length 12 units as shown in Figure 3.
3. Find the coordinates of \(A\) and the coordinates of \(B\).
4. Show that angle \(A N B = 134.8 ^ { \circ }\), to the nearest 0.1 of a degree. The tangents to \(C\) at the points \(A\) and \(B\) meet at the point \(P\).
5. Find the length \(A P\), giving your answer to 3 significant figures.

9. The points \(A\) and \(B\) have coordinates \(( - 2,11 )\) and \(( 8,1 )\) respectively. Given that \(A B\) is a diameter of the circle \(C\),

1. show that the centre of \(C\) has coordinates \(( 3,6 )\),
2. find an equation for \(C\).
3. Verify that the point \(( 10,7 )\) lies on \(C\).
4. Find an equation of the tangent to \(C\) at the point (10, 7), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

5. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 24 y + 195 = 0$$ The centre of \(C\) is at the point \(M\).

1. Find
   1. the coordinates of the point \(M\),
   2. the radius of the circle \(C\). \(N\) is the point with coordinates \(( 25,32 )\).
2. Find the length of the line \(M N\). The tangent to \(C\) at a point \(P\) on the circle passes through point \(N\).
3. Find the length of the line \(N P\).

8. The circle \(C\), with centre at the point \(A\), has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0\). Find

1. the coordinates of \(A\),
2. the radius of \(C\),
3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. Given that the line \(l\) with gradient \(\frac { 7 } { 2 }\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),
4. find an equation of the line which passes through \(A\) and \(T\).

7. \begin{figure}[h] \end{figure} The line \(y = 3 x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(P ( 2,2 )\), as shown in Figure 1. The point \(Q\) is the centre of \(C\).

1. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
2. show that the \(x\)-coordinate of \(Q\) is 5,
3. find an equation for \(C\).

7. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3.
The point \(A\) has coordinates \(( 1 , - 2 )\) and the mid-point \(M\) of \(A B\) has coordinates \(( 3,1 )\). The line \(l\) passes through the points \(M\) and \(P\).

1. Find an equation for \(l\). Given that the \(x\)-coordinate of \(P\) is 6 ,
2. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is - 1 ,
3. find an equation for the circle.