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

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.

1. The circle \(C\)

• has centre \(A ( 3,5 )\)
• passes through the point \(B ( 8 , - 7 )\)
  1. Find an equation for \(C\).

The points \(M\) and \(N\) lie on \(C\) such that \(M N\) is a chord of \(C\).
Given that \(M N\)

• lies above the \(x\)-axis
• is parallel to the \(x\)-axis
• has length \(4 \sqrt { 22 }\)
• find an equation for the line passing through points \(M\) and \(N\).

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.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a circle \(C\) with centre \(N ( 4 , - 1 )\). The line \(l\) with equation \(y = \frac { 1 } { 2 } x\) is a tangent to \(C\) at the point \(P\). Find

1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
2. the equation of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{bfeb1724-9a00-4a36-9606-520395792b2b-16_2256_52_311_1978}

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

1. The curve \(C\) has equation

$$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$

1. Write the equation of \(C\) in the form $$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$ where \(a , b , c\) and \(d\) are fully simplified constants. The curve \(C\) has three turning points.
   Using calculus,
2. show that the \(x\) coordinates of the three turning points satisfy the equation $$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$ Given that the \(x\) coordinate of one of the turning points is 1
3. find, using algebra, the exact \(x\) coordinates of the other two turning points.
   (Solutions based entirely on calculator technology are not acceptable.)

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 3 shows

• the curve \(C _ { 1 }\) with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 3 x + 14\)
• the circle \(C _ { 2 }\) with centre \(T\)

The point \(T\) is the minimum turning point of \(C _ { 1 }\) Using Figure 3 and calculus,

1. find the coordinates of \(T\) The curve \(C _ { 1 }\) intersects the circle \(C _ { 2 }\) at the point \(A\) with \(x\) coordinate 2
2. Find an equation of the circle \(C _ { 2 }\) The line \(l\) shown in Figure 3, is the tangent to circle \(C _ { 2 }\) at \(A\)
3. Show that an equation of \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 22 } { 3 }$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C _ { 1 } , l\) and the \(y\)-axis.
4. Find the exact area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of

• the circle \(C\) with centre \(X ( 4 , - 3 )\)
• the line \(l\) with equation \(y = \frac { 5 } { 2 } x - \frac { 55 } { 2 }\)

Given that \(l\) is the tangent to \(C\) at the point \(N\),

1. show that an equation for the straight line passing through \(X\) and \(N\) is $$2 x + 5 y + 7 = 0$$
2. Hence find
   1. the coordinates of \(N\),
   2. an equation for \(C\).

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 \(C\).

3. The line joining the points \(( - 1,4 )\) and \(( 3,6 )\) is a diameter of the circle \(C\). Find an equation for \(C\).

5. \begin{figure}[h] \end{figure} The points \(P ( - 3,2 ) , Q ( 9,10 )\) and \(R ( a , 4 )\) lie on the circle \(C\), as shown in Figure 2. Given that \(P R\) is a diameter of \(C\),

1. show that \(a = 13\),
2. find an equation for \(C\).

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.

2. A circle \(C\) has centre \(( - 1,7 )\) and passes through the point \(( 0,0 )\). Find an equation for \(C\).
(4)

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. \begin{figure}[h] \end{figure} Figure 2 shows a circle \(C\) with centre \(O\) and radius 5

1. Write down the cartesian equation of \(C\). The points \(P ( - 3 , - 4 )\) and \(Q ( 3 , - 4 )\) lie on \(C\).
2. Show that the tangent to \(C\) at the point \(Q\) has equation $$3 x - 4 y = 25$$
3. Show that, to 3 decimal places, angle \(P O Q\) is 1.287 radians. The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) intersect on the \(y\)-axis at the point \(R\).
4. Find the area of the shaded region \(P Q R\) shown in Figure 2. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-25_177_154_2576_1804}

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.