Circle from diameter endpoints

5 A circle has equation \(( x - 1 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 40\). A line with equation \(y = x - 9\) intersects the circle at points \(A\) and \(B\).

1. Find the coordinates of the two points of intersection.
2. Find an equation of the circle with diameter \(A B\).

12 A diameter of a circle \(C _ { 1 }\) has end-points at \(( - 3 , - 5 )\) and \(( 7,3 )\).

1. Find an equation of the circle \(C _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-16_618_846_1062_644} The circle \(C _ { 1 }\) is translated by \(\binom { 8 } { 4 }\) to give circle \(C _ { 2 }\), as shown in the diagram.
2. Find an equation of the circle \(C _ { 2 }\).
   The two circles intersect at points \(R\) and \(S\).
3. Show that the equation of the line \(R S\) is \(y = - 2 x + 13\).
4. Hence show that the \(x\)-coordinates of \(R\) and \(S\) satisfy the equation \(5 x ^ { 2 } - 60 x + 159 = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The circle with equation \(( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 40\) intersects the \(y\)-axis at points \(A\) and \(B\).

1. Find the \(y\)-coordinates of \(A\) and \(B\), expressing your answers in terms of surds.
2. Find the equation of the circle which has \(A B\) as its diameter.

15. The points \(A\) and \(B\) have coordinates \(( - 8 , - 8 )\) and \(( 12,2 )\) respectively. \(A B\) is the diameter of a circle \(C\).

1. Find an equation for the circle \(C\). The point \(( 4,8 )\) also lies on \(C\).
2. Find an equation of the tangent to \(C\) at the point ( 4,8 ), giving your answer in the form \(a x + b y + c = 0\)

4. The points \(P\) and \(Q\) have coordinates \(( - 11,6 )\) and \(( - 3,12 )\) respectively. Given that \(P Q\) is a diameter of the circle \(C\),

1. 1. find the coordinates of the centre of \(C\),
   2. find the radius of \(C\).
2. Hence find an equation of \(C\).
3. Find an equation of the tangent to \(C\) at the point \(Q\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{0e107b51-2fb3-4ad7-8542-5aa0da13b127-13_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

6. (i) The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 12 y = k \quad \text { where } k \text { is a constant }$$

1. Find the coordinates of the centre of \(C _ { 1 }\)
2. State the possible range in values for \(k\).
   (ii) The point \(P ( p , 0 )\), the point \(Q ( - 2,10 )\) and the point \(R ( 8 , - 14 )\) lie on a different circle, \(C _ { 2 }\) Given that

3. \begin{figure}[h] \end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find

1. the exact length of \(A B\),
2. the coordinates of the midpoint \(P\) of \(A B\),
3. an equation for the circle \(C\).

3. \begin{figure}[h] \end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find

1. the exact length of \(A B\),
2. the coordinates of the midpoint \(P\) of \(A B\),
3. an equation for the circle \(C\).

3. The line joining the points \(( - 1,4 )\) and \(( 3,6 )\) is a diameter of the circle \(C\). Find an equation for \(C\).

5. \begin{figure}[h] \end{figure} The points \(P ( - 3,2 ) , Q ( 9,10 )\) and \(R ( a , 4 )\) lie on the circle \(C\), as shown in Figure 2. Given that \(P R\) is a diameter of \(C\),

1. show that \(a = 13\),
2. find an equation for \(C\).

9. The points \(A\) and \(B\) have coordinates \(( - 2,11 )\) and \(( 8,1 )\) respectively. Given that \(A B\) is a diameter of the circle \(C\),

1. show that the centre of \(C\) has coordinates \(( 3,6 )\),
2. find an equation for \(C\).
3. Verify that the point \(( 10,7 )\) lies on \(C\).
4. Find an equation of the tangent to \(C\) at the point (10, 7), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

1. The circle \(C\), with centre \(A\), passes through the point \(P\) with coordinates ( \(- 9,8\) ) and the point \(Q\) with coordinates \(( 15 , - 10 )\).

Given that \(P Q\) is a diameter of the circle \(C\),

1. find the coordinates of \(A\),
2. find an equation for \(C\). A point \(R\) also lies on the circle \(C\).
   Given that the length of the chord \(P R\) is 20 units,
3. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
4. Find the size of the angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.

The points \(A\) and \(B\) have coordinates \(( 5 , - 1 )\) and \(( 13,11 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\). Given that \(A B\) is a diameter of the circle \(C\),
2. find an equation for \(C\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c532661c-8a94-483a-a921-b35d5c0a0188-04_754_810_1053_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.

1. Calculate the radius of the circle and the coordinates of the centre.
2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).

7 \includegraphics[max width=\textwidth, alt={}, center]{5fa27228-37b2-45d9-a8dc-355b2f7f6fa4-3_757_810_1050_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.

1. Calculate the radius of the circle and the coordinates of the centre.
2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).

2 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 \(y = a x + b\).

6 The points \(\mathrm { A } ( - 1,6 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 13,4 )\) 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.

3 Find the coordinates of the point of intersection of the lines \(y = 3 x + 1\) and \(x + 3 y = 6\). \begin{figure}[h] \end{figure} The line AB has equation \(y = 4 x - 5\) and passes through the point \(\mathrm { B } ( 2,3 )\), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C . Find the equation of the line BC and the \(x\)-coordinate of C . \(5 \mathrm {~A} ( 9,8 ) , \mathrm { B } ( 5,0 )\) an \(\mathrm { C } ( 3,1 )\) are three points.

1. Show that AB and BC are perpendicular.
2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle.
3. BD is a diameter of the circle. Find the coordinates of D .

7 The line with equation \(3 x + 4 y - 10 = 0\) passes through point \(A ( 2,1 )\) and point \(B ( 10 , k )\).

1. Find the value of \(k\).
2. Calculate the length of \(A B\). A circle has equation \(( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
3. Write down the coordinates of the centre and the radius of the circle.
4. Verify that \(A B\) is a diameter of the circle.

9

1. The line joining the points \(A ( 4,5 )\) and \(B ( p , q )\) has mid-point \(M ( - 1,3 )\). Find \(p\) and \(q\). \(A B\) is the diameter of a circle.
2. Find the radius of the circle.
3. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
4. Find an equation of the tangent to the circle at the point \(( 4,5 )\).

9 The points \(A ( 1,3 ) , B ( 7,1 )\) and \(C ( - 3 , - 9 )\) are joined to form a triangle.

1. Show that this triangle is right-angled and state whether the right angle is at \(A , B\) or \(C\).
2. The points \(A , B\) and \(C\) lie on the circumference of a circle. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).

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.