1.03f Circle properties: angles, chords, tangents

9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 2 y - 3 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle.
2. Find the values of \(k\) for which the line \(y = k\) is a tangent to the circle, giving your answers in simplified surd form.
3. The points \(S\) and \(T\) lie on the circumference of the circle. \(M\) is the mid-point of the chord \(S T\). Given that the length of \(C M\) is 2 , calculate the length of the chord \(S T\).
4. Find the coordinates of the point where the circle meets the line \(x - 2 y - 12 = 0\).

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 \(( 4,5 )\).

10 \includegraphics[max width=\textwidth, alt={}, center]{0ae3af7e-32cc-43fa-89bb-d6697a8f8061-3_755_905_248_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 6 y - 20 = 0\).

1. Find the centre and radius of the circle. The circle crosses the positive \(x\)-axis at the point \(A\).
2. Find the equation of the tangent to the circle at \(A\).
3. A second tangent to the circle is parallel to the tangent at \(A\). Find the equation of this second tangent.
4. Another circle has centre at the origin \(O\) and radius \(r\). This circle lies wholly inside the first circle. Find the set of possible values of \(r\).

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.

1. (i) A circle \(C _ { 1 }\) has equation

$$x ^ { 2 } + y ^ { 2 } + 18 x - 2 y + 30 = 0$$ The line \(l\) is the tangent to \(C _ { 1 }\) at the point \(P ( - 5,7 )\).
Find an equation of \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
(ii) A different circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 12 y + k = 0$$ where \(k\) is a constant.
Given that \(C _ { 2 }\) lies entirely in the 4th quadrant, find the range of possible values for \(k\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the circle \(C\)

• the point \(P ( - 1 , k + 8 )\) is the centre of \(C\)
• the point \(Q \left( 3 , k ^ { 2 } - 2 k \right)\) lies on \(C\)
• \(k\) is a positive constant
• the line \(l\) is the tangent to \(C\) at \(Q\)

Given that the gradient of \(l\) is - 2

1. show that $$k ^ { 2 } - 3 k - 10 = 0$$
2. Hence find an equation for \(C\)

6 \includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-4_442_661_840_260} The diagram shows triangle \(A B C\), with \(A B = x \mathrm {~cm} , A C = y \mathrm {~cm}\) and angle \(B A C = 60 ^ { \circ }\). It is given that the area of the triangle is \(( x + y ) \sqrt { 3 } \mathrm {~cm} ^ { 2 }\).

1. Show that \(4 x + 4 y = x y\). When the vertices of the triangle are placed on the circumference of a circle, \(A C\) is a diameter of the circle.
2. Determine the value of \(x\) and the value of \(y\).

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 .

14

1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
2. Prove that \(h = \sqrt { a b }\), as given in line 24 .

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 circle with centre \(C ( 5,8 )\) touches the \(y\)-axis, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-5_485_631_370_715}

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. 1. Verify that the point \(A ( 2,12 )\) lies on the circle.
   2. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(s x + t y + u = 0\), where \(s , t\) and \(u\) are integers.
3. The points \(P\) and \(Q\) lie on the circle, and the mid-point of \(P Q\) is \(M ( 7,12 )\).
   1. Show that the length of \(C M\) is \(n \sqrt { 5 }\), where \(n\) is an integer.
   2. Hence find the area of triangle \(P C Q\).

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. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 8 y + k = 0 ,$$ where \(k\) is a constant. Given that the point with coordinates \(( - 6,5 )\) lies on \(C\),

1. find the value of \(k\),
2. find the coordinates of the centre and the radius of \(C\). A straight line which passes through the point \(A ( 2,3 )\) is a tangent to \(C\) at the point \(B\).
3. Find the length \(A B\) in the form \(k \sqrt { 3 }\).

7. The circle \(C\) has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).

1. Find the length of the diameter of \(C\).
2. Find an equation for \(C\).
3. Show that the line \(y = 2 x - 3\) is a tangent to \(C\) and find the coordinates of the point of contact.

6 In this question you must show detailed reasoning. The diagram shows the line \(3 y + x = 7\) which is a tangent to a circle with centre \(( 3 , - 2 )\).
Find an equation for the circle.

5 A circle with centre \(C\) has equation \(( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25\).

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the point \(N ( 0 , - 2 )\) lies on the circle.
   2. Sketch the circle.
   3. Find an equation of the normal to the circle at the point \(N\).
3. The point \(P\) has coordinates (2, 6).
   1. Find the distance \(P C\), leaving your answer in surd form.
   2. Find the length of a tangent drawn from \(P\) to the circle.