Circle from diameter endpoints

11 A circle has equation \(( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).

1. State the coordinates of the centre of this circle and its radius.
2. Verify that the point A with coordinates \(( 6 , - 6 )\) lies on this circle. Show also that the point B on the circle for which AB is a diameter has coordinates \(( 0,2 )\).
3. Find the equation of the tangent to the circle at A .
4. A second circle touches the original circle at A . Its radius is 10 and its centre is at C , where BAC is a straight line. Find the coordinates of C and hence write down the equation of this second circle.

11 The points \(A ( - 1,6 ) , B ( 1,0 )\) and \(C ( 13,4 )\) are joined by straight lines.

1. Prove that the lines AB and BC are perpendicular.
2. Find the area of triangle ABC .
3. A circle passes through the points A , B and C . Justify the statement that AC is a diameter of this circle. Find the equation of this circle.
4. Find the coordinates of the point on this circle that is furthest from B .

10 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h] \end{figure}

1. Write down the coordinates of B .
2. Find the radius of the circle and hence write down the equation of the circle.
3. AD is a diameter of the circle. Find the coordinates of D .
4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-4_643_853_269_589} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
5. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
6. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
7. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
8. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-5_775_1461_317_296} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).
9. Use the intersections with both axes to express the equation of the curve in a factorised form.
10. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
11. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
12. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0 .$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. \section*{END OF QUESTION PAPER}

10 Fig. 10 shows a sketch of the points \(\mathrm { A } ( 2,7 ) , \mathrm { B } ( 0,3 )\) and \(\mathrm { C } ( 8 , - 1 )\). \begin{figure}[h] \end{figure}

1. Prove that angle ABC is \(90 ^ { \circ }\).
2. Find the equation of the circle which has AC as a diameter.
3. Find the equation of the tangent to this circle at A . Give your answer in the form \(a y = b x + c\), where \(a , b\) and \(c\) are integers.

3 The points \(P\) and \(Q\) have coordinates \(( 2 , - 5 )\) and \(( 3,1 )\) respectively.
Determine the equation of the circle that has \(P Q\) as a diameter. Give your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5 A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
4. The point \(T\) lies on the tangent to the circle at \(A\) such that \(A T = 4\). Find the length of \(C T\).

7. \begin{figure}[h] \end{figure} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 1.

1. Find the coordinates of the centre of the circle.
2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.

7. \begin{figure}[h] \end{figure} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 1.

1. Find the coordinates of the centre of the circle.
2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.

2. The point \(A\) has coordinates \(( 2,5 )\) and the point \(B\) has coordinates \(( - 2,8 )\). Find, in cartesian form, an equation of the circle with diameter \(A B\).

7. The points \(P\) and \(Q\) have coordinates \(( - 2,6 )\) and \(( 4 , - 1 )\) respectively. Given that \(P Q\) is a diameter of circle \(C\),

1. find the coordinates of the centre of \(C\),
2. show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0 .$$ The point \(R\) has coordinates (2, 7).
3. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle P R Q\) in degrees.

16. \section*{Figure 3} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.

1. Find the coordinates of the centre of the circle.
2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.

16. \section*{Figure 3} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.

1. Find the coordinates of the centre of the circle.
2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.

5 Express \(3 x ^ { 3 } + 5 x ^ { 2 } - 27 x + 10\) in the form \(( x - 2 ) \left( a x ^ { 2 } + b x + c \right)\), where \(a , b\) and \(c\) are integers.
[0pt] [3 marks]
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-07_2488_1716_219_153}
\(6 \quad A B\) is a diameter of a circle where \(A\) is \(( 1,4 )\) and \(B\) is \(( 7 , - 2 )\)

6

1. Find the coordinates of the midpoint of \(A B\). 6
2. Show that the equation of the circle may be written as $$x ^ { 2 } + y ^ { 2 } - 8 x - 2 y = 1$$ 6
3. \(\quad\) The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\). Find the exact area of triangle \(D E C\). 6
4. The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\).
   The area enclosed between the curve and the \(x\)-axis is 36 units.
   Find the value of \(a\).
   Fully justify your answer.
   [0pt] [6 marks]
   \(7 \quad\) A curve has equation \(y = a ^ { 2 } - x ^ { 2 }\), where \(a > 0\)

7 The points \(A ( a , 3 )\) and \(B ( 10,6 )\) lie on a circle.
\(A B\) is a diameter of the circle and passes through the point ( 2,4 )
The circle has equation $$( x - c ) ^ { 2 } + ( y - d ) ^ { 2 } = e$$ where \(c , d\) and \(e\) are rational numbers. Find the values of \(a , c , d\) and \(e\).