Point position relative to circle

1. The circle \(C\) has equation

$$( x - 3 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 30$$ Write down

1. 1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). Given that the point \(P\) with coordinates \(( 6 , k )\), where \(k\) is a constant, lies inside circle \(C\), (b) show that $$k ^ { 2 } + 8 k - 5 < 0$$
2. Hence find the exact set of values of \(k\) for which \(P\) lies inside \(C\).
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1．A point \(P\) lies on the curve with equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y = 24$$ Find the greatest and least possible values of the length \(O P\) ，where \(O\) is the origin．

10 A circle has equation \(( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).

1. Find the coordinates of the centre \(C\) and the length of the diameter.
2. Find the equation of the line which passes through \(C\) and the point \(P ( 7,2 )\).
3. Calculate the length of \(C P\) and hence determine whether \(P\) lies inside or outside the circle.
4. Determine algebraically whether the line with equation \(y = 2 x\) meets the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

9 A circle with centre \(C\) has equation \(( x - 2 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 25\).

1. Show that no part of the circle lies above the \(x\)-axis.
2. The point \(P\) has coordinates \(( 6 , k )\) and lies inside the circle. Find the set of possible values of \(k\).
3. Prove that the line \(2 y = x\) does not meet the circle.

1. A circle has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 16 y = 80$$

1. Find
   1. the coordinates of the centre of the circle,
   2. the radius of the circle. Given that \(P\) is the point on the circle that is furthest away from the origin \(O\),
2. find the exact length \(O P\)

5 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x + 6 y = 11\).

1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. The point \(O\) has coordinates \(( 0,0 )\).
   1. Find the length of CO .
   2. Hence determine whether the point \(O\) lies inside or outside the circle, giving a reason for your answer.

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 4 x + 12 y + 15 = 0\).

1. Find:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. Explain why the circle lies entirely below the \(x\)-axis.
3. The point \(P\) with coordinates \(( 5 , k )\) lies outside the circle.
   1. Show that \(P C ^ { 2 } = k ^ { 2 } + 12 k + 45\).
   2. Hence show that \(k ^ { 2 } + 12 k + 20 > 0\).
   3. Find the possible values of \(k\).

11

1. A circle has equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0$$ Find the centre and the radius of the circle.
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2. An equilateral triangle has one vertex at the origin, and one side along the line \(x = 8\), as shown in the diagram below.
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3. 1. Show that the vertex at the origin lies inside the circle \(x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0\)
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4. (ii) Prove that the triangle lies completely within the circle \(x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0\)