Geometric properties with circles

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\includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-18_750_981_258_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y - 27 = 0\) and the tangent to the circle at the point \(P ( 5,4 )\).

1. The tangent to the circle at \(P\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Find the area of triangle \(O A B\), where \(O\) is the origin.
2. Points \(Q\) and \(R\) also lie on the circle, such that \(P Q R\) is an equilateral triangle. Find the exact area of triangle \(P Q R\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(R\).

4. \begin{figure}[h] \end{figure} Figure 1 shows the equilateral triangle \(L M N\) of side 2 cm ．The point \(P\) lies on \(L M\) such that \(L P = x \mathrm {~cm}\) and the point \(Q\) lies on \(L N\) such that \(L Q = y \mathrm {~cm}\) ．The points \(P\) and \(Q\) are chosen so that the area of triangle \(L P Q\) is half the area of triangle \(L M N\) ．
（a）Show that \(x y = 2\)
（b）Find the shortest possible length of \(P Q\) ，justifying your answer． Mathematicians know that for any closed curve or polygon enclosing a fixed area，the ratio \(\frac { \text { area enclosed } } { \text { perimeter } }\) is a maximum when the closed curve is a circle． By considering 6 copies of triangle \(L M N\) suitably arranged，
（c）find the length of the shortest line or curve that can be drawn from a point on \(L M\) to a point on \(L N\) to divide the area of triangle \(L M N\) in half．Justify your answer．
（6）

12 Fig. 12 shows the circle \(( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 25\), the line \(4 y = 3 x - 32\) and the tangent to the circle at the point \(\mathrm { A } ( 5,2 )\). D is the point of intersection of the line \(4 y = 3 x - 32\) and the tangent at A . \begin{figure}[h] \end{figure}

1. Write down the coordinates of C , the centre of the circle.
2. (A) Show that the line \(4 y = 3 x - 32\) is a tangent to the circle.
   (B) Find the coordinates of B , the point where the line \(4 y = 3 x - 32\) touches the circle.
3. Prove that ADBC is a square.
4. The point E is the lowest point on the circle. Find the area of the sector ECB .

15 The circle \(x ^ { 2 } + y ^ { 2 } + 2 x - 14 y + 25 = 0\) has its centre at the point \(C\). The line \(7 y = x + 25\) intersects the circle at points A and B . Prove that triangle ABC is a right-angled triangle.

8 A circle with centre \(A\) and radius 8 cm and a circle with centre \(C\) and radius 12 cm intersect at points B and D . Quadrilateral \(A B C D\) has area \(60 \mathrm {~cm} ^ { 2 }\).
Determine the two possible values for the length AC.

6 The diagram shows a circle \(C\) and a line \(L\), which is the tangent to \(C\) at the point \(( 1,1 )\). The equations of \(C\) and \(L\) are $$x ^ { 2 } + y ^ { 2 } = 2 \text { and } x + y = 2$$ respectively.
\includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_760_1301_552_395} The circle \(C\) is now transformed by a stretch with scale factor 2 parallel to the \(x\)-axis. The image of \(C\) under this stretch is an ellipse \(E\).

1. On the diagram below, sketch the ellipse \(E\), indicating the coordinates of the points where it intersects the coordinate axes.
2. Find equations of:
   1. the ellipse \(E\);
   2. the tangent to \(E\) at the point \(( 2,1 )\).
      \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_743_1301_1921_420}

11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).

1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.

4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).

1. Use the graph to solve the following equations, showing your method clearly.
   (A) \(x + \frac { 1 } { x } = 4\)
   (B) \(2 x + \frac { 1 } { x } = 4\)
2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.

6 A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(A B\) is \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{40d40a0b-5b33-4940-b15b-ee03e1291f61-06_794_753_351_239} The equation of the circle is \(x ^ { 2 } - 6 x + y ^ { 2 } + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Show that the area of triangle \(A B C\) is \(\frac { 3 } { 2 } \sqrt { 9 - 2 a }\).
2. 1. Find the value of \(a\) when the area of triangle \(A B C\) is zero.
   2. Give a geometrical interpretation of the case in part (b)(i).
3. Give a geometrical interpretation of the case where \(a = 5\).

7 Three points \(A , B\) and \(C\) have coordinates \(A ( 8,17 ) , B ( 15,10 )\) and \(C ( - 2 , - 7 )\) 7

1. Show that angle \(A B C\) is a right angle.
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2. \(\quad A , B\) and \(C\) lie on a circle.
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3. 1. Explain why \(A C\) is a diameter of the circle.
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4. (ii) Determine whether the point \(D ( - 8 , - 2 )\) lies inside the circle, on the circle or outside the circle. Fully justify your answer.

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1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where \(a , b\) and \(k\) are constants to be found.
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2. State the coordinates of \(P\) 9
3. Find the \(y\)-coordinate of \(Q\)
   \section*{Question 9 continues on the next page} 9
4. The line segment \(Q R\) is a tangent to the circle as shown in Figure 2 below. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-18_885_1180_456_495}
   \end{figure} The point \(R\) has coordinates \(( 9 , - 3 )\).
   Find the angle QPR
   Give your answer in radians to three significant figures.
   It is given that $$f ( x ) = 5 x ^ { 3 } + x$$ Use differentiation from first principles to prove that $$f ^ { \prime } ( x ) = 15 x ^ { 2 } + 1$$

14 Curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$$ 14

1. Curve \(C _ { 2 }\) is a reflection of \(C _ { 1 }\) in the line \(y = x\)
   Write down an equation of \(C _ { 2 }\)
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2. Curve \(C _ { 3 }\) is a circle of radius 4 , centred at the origin.
   Describe a single transformation which maps \(C _ { 1 }\) onto \(C _ { 3 }\)
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3. Curve \(C _ { 4 }\) is a translation of \(C _ { 1 }\)
   The positive \(x\)-axis and the positive \(y\)-axis are tangents to \(C _ { 4 }\)
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4. 1. Sketch the graphs of \(C _ { 1 }\) and \(C _ { 4 }\) on the axes opposite. Indicate the coordinates of the \(x\) and \(y\) intercepts on your graphs.
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5. (ii) Determine the translation vector.
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6. (iii) The line \(y = m x + c\) is a tangent to both \(C _ { 1 }\) and \(C _ { 4 }\) Find the value of \(m\)