1.03f Circle properties: angles, chords, tangents

7 \includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-10_716_899_258_621} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters \(C A D\) and \(B A E\) are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).

1. Show that the radius of the larger circle is \(r \sqrt { } 2\).
2. Find the area of the shaded region in terms of \(r\).

6 \includegraphics[max width=\textwidth, alt={}, center]{cc85b13a-7f15-4025-a545-373cda454de8-3_456_495_255_824} The diagram shows a circle with centre \(O\) and radius 10 cm . The chord \(A B\) divides the circle into two regions whose areas are in the ratio \(1 : 4\) and it is required to find the length of \(A B\). The angle \(A O B\) is \(\theta\) radians.

1. Show that \(\theta = \frac { 2 } { 5 } \pi + \sin \theta\).
2. Showing all your working, use an iterative formula, based on the equation in part (i), with an initial value of 2.1 , to find \(\theta\) correct to 2 decimal places. Hence find the length of \(A B\) in centimetres correct to 1 decimal place.

12. Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\). \(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find

1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
2. the perimeter of the platform to 3 significant figures,
3. the area of the platform to 3 significant figures.

9. \begin{figure}[h] \end{figure} In Figure 3, the points \(A\) and \(B\) are the centres of the circles \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The circle \(C _ { 1 }\) has radius 10 cm and the circle \(C _ { 2 }\) has radius 5 cm . The circles intersect at the points \(X\) and \(Y\), as shown in the figure. Given that the distance between the centres of the circles is 12 cm ,

1. calculate the size of the acute angle \(X A B\), giving your answer in radians to 3 significant figures,
2. find the area of the major sector of circle \(C _ { 1 }\), shown shaded in Figure 3,
3. find the area of the kite \(A Y B X\).

11. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x - 10 y + 16 = 0$$ The centre of \(C\) is at the point \(T\).

1. Find
   1. the coordinates of the point \(T\),
   2. the radius of the circle \(C\). The point \(M\) has coordinates \(( 20,12 )\).
2. Find the exact length of the line \(M T\). Point \(P\) lies on the circle \(C\) such that the tangent at \(P\) passes through the point \(M\).
3. Find the exact area of triangle \(M T P\), giving your answer as a simplified surd.

9. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 9 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\). The point \(P ( - 2,7 )\) lies on \(C\).
3. Find an equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   

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 \(N\).

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.

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

9. \begin{figure}[h] \end{figure} Figure 2 shows a plan of a patio. The patio \(P Q R S\) is in the shape of a sector of a circle with centre \(Q\) and radius 6 m . Given that the length of the straight line \(P R\) is \(6 \sqrt { } 3 \mathrm {~m}\),

1. find the exact size of angle \(P Q R\) in radians.
2. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
3. Find the exact area of the triangle \(P Q R\).
4. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment \(P R S\).
5. Find, in \(m\) to 1 decimal place, the perimeter of the patio \(P Q R S\).

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.

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}

6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y = 12$$

1. Find the centre and the radius of \(C\). The point \(P ( - 1,1 )\) and the point \(Q ( 7 , - 5 )\) both lie on \(C\).
2. Show that \(P Q\) is a diameter of \(C\). The point \(R\) lies on the positive \(y\)-axis and the angle \(P R Q = 90 ^ { \circ }\).
3. Find the coordinates of \(R\).

10. \begin{figure}[h] \end{figure} The circle \(C\) has radius 5 and touches the \(y\)-axis at the point \(( 0,9 )\), as shown in Figure 4.

1. Write down an equation for the circle \(C\), that is shown in Figure 4. A line through the point \(P ( 8 , - 7 )\) is a tangent to the circle \(C\) at the point \(T\).
2. Find the length of \(P T\).

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

Given that \(P Q\) is a diameter of the circle \(C\),

1. find the coordinates of \(A\),
2. find an equation for \(C\). A point \(R\) also lies on the circle \(C\).
   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 the angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.

9. \begin{figure}[h] \end{figure} Figure 3 shows a circle \(C\) with centre \(Q\) and radius 4 and the point \(T\) which lies on \(C\). The tangent to \(C\) at the point \(T\) passes through the origin \(O\) and \(O T = 6 \sqrt { } 5\) Given that the coordinates of \(Q\) are \(( 11 , k )\), where \(k\) is a positive constant, (a) find the exact value of \(k\),
(b) find an equation for \(C\).

2. A circle \(C\) with centre at the point \(( 2 , - 1 )\) passes through the point \(A\) at \(( 4 , - 5 )\).

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

4. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 12 x\) The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on \(C\), where \(p \neq 0\)

1. Show that the equation of the normal to the curve \(C\) at the point \(P\) is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve \(C\) again at the point \(Q\).
   Given that \(p = 2\) and that \(S\) is the focus of the parabola, find
2. the coordinates of the point \(Q\),
3. the area of the triangle \(P Q S\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 36 x\)

The point \(P \left( 9 t ^ { 2 } , 18 t \right)\), where \(t \neq 0\), lies on \(C\)

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$y + t x = 9 t ^ { 3 } + 18 t$$
2. Hence find the equations of the two normals to \(C\) which pass through the point (54, 0), giving your answers in the form \(y = p x + q\) where \(p\) and \(q\) are constants to be determined. Given that
   • the normals found in part (b) intersect the directrix of \(C\) at the points \(A\) and \(B\)
   • the point \(F\) is the focus of \(C\)
   • determine the area of triangle \(A F B\)

7 \includegraphics[max width=\textwidth, alt={}, center]{c532661c-8a94-483a-a921-b35d5c0a0188-04_754_810_1053_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.

1. Calculate the radius of the circle and the coordinates of the centre.
2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).

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 \(y = m x + c\).