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 \(r\).

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.

Circles \(C_1\) and \(C_2\) have equations $$x^2 + y^2 + 6x - 10y + 18 = 0 \text{ and } (x-9)^2 + (y+4)^2 - 64 = 0$$ respectively.

1. Find the distance between the centres of the circles. [4] \(P\) 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 \(d\). [3]

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

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). [3]

A second circle has centre at the point \((15, 12)\) and radius \(10\).

1. Sketch both circles on a single diagram and find the coordinates of the point where they touch. [4]

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

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). [3]

A second circle has centre at the point \((15, 12)\) and radius \(10\).

1. Sketch both circles on a single diagram and find the coordinates of the point where they touch. [4]

A circle, C, has equation \(x^2 - 6x + y^2 = 16\). A second circle, D, has the following properties:

• The line through the centres of circle C and circle D has gradient 1.
• Circle D touches circle C at exactly one point.
• The centre of circle D lies in the first quadrant.
• Circle D has the same radius as circle C.

Find the coordinates of the centre of circle D. [5]

The circles \(C_1\) and \(C_2\) have respective equations $$x^2 + y^2 - 6x - 2y = 15$$ $$x^2 + y^2 - 18x + 14y = 95.$$

1. By considering the coordinates of the centres and the lengths of the radii of \(C_1\) and \(C_2\), show that \(C_1\) and \(C_2\) touch internally at some point \(P\). [4]
2. Determine the coordinates of \(P\). [3]
3. Find the equation of the common tangent to the circles at P. [4]