Two circles intersection or tangency

8
\includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-10_492_888_255_625} The diagram shows two identical circles intersecting at points \(A\) and \(B\) and with centres at \(P\) and \(Q\). The radius of each circle is \(r\) and the distance \(P Q\) is \(\frac { 5 } { 3 } r\).

1. Find the perimeter of the shaded region in terms of \(r\).
2. Find the area of the shaded region in terms of \(r\).

6 Circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 18 = 0 \text { and } ( x - 9 ) ^ { 2 } + ( y + 4 ) ^ { 2 } - 64 = 0$$ respectively.

1. Find the distance between the centres of the circles.
   \(P\) and \(Q\) are points on \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The distance between \(P\) and \(Q\) is denoted by \(d\).
2. Find the greatest and least possible values of \(d\).

7
\includegraphics[max width=\textwidth, alt={}, center]{4782d612-0ec1-418e-8ef3-c871dce82b44-10_611_732_255_705} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(C A D\) is a straight line and \(A B\) is perpendicular to \(C D\).

1. Find angle \(A B C\) in radians.
2. Find the area of the shaded region.
   \(8 \quad A ( - 1,1 )\) and \(P ( a , b )\) are two points, where \(a\) and \(b\) are constants. The gradient of \(A P\) is 2 .
3. Find an expression for \(b\) in terms of \(a\).
4. \(B ( 10 , - 1 )\) is a third point such that \(A P = A B\). Calculate the coordinates of the possible positions of \(P\).

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 \(C _ { 3 }\)

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.

11. \begin{figure}[h] \end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 100\)
Circle \(C _ { 2 }\) has equation \(( x - 15 ) ^ { 2 } + y ^ { 2 } = 40\)
The circles meet at points \(A\) and \(B\) as shown in Figure 3.

1. Show that angle \(A O B = 0.635\) radians to 3 significant figures, where \(O\) is the origin. The region shown shaded in Figure 3 is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
2. Find the perimeter of the shaded region, giving your answer to one decimal place.

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 \(C _ { 2 }\)
      • has centre with coordinates (-6, -8)
2. has radius \(k\), where \(k\) is a constant
Given that \(C _ { 1 }\) and \(C _ { 2 }\) intersect at 2 distinct points,
3. find the range of values of \(k\), writing your answer in set notation.

3. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y - 75 = 0$$

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). A second circle has centre at the point \(( 15,12 )\) and radius 10 .
2. Sketch both circles on a single diagram and find the coordinates of the point where they touch.
   (4)