1.03e Complete the square: find centre and radius of circle

5. The circle \(C\) has centre \(( 3,1 )\) and passes through the point \(P ( 8,3 )\).

1. Find an equation for \(C\).
2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 11 = 0$$ Find

1. the coordinates of the centre of \(C\),
2. the radius of \(C\),
3. the coordinates of the points where \(C\) crosses the \(y\)-axis, giving your answers as simplified surds.

3. \begin{figure}[h] \end{figure} The circle \(C\) with centre \(T\) and radius \(r\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Show that \(r = 5\) The line \(L\) has equation \(x = 13\) and crosses \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.
3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). Given that, to 3 decimal places, the angle \(P T Q\) is 1.855 radians,
4. find the perimeter of the sector \(P T Q\).

10. \begin{figure}[h] \end{figure} The circle \(C\) has radius 5 and touches the \(y\)-axis at the point \(( 0,9 )\), as shown in Figure 4.

1. Write down an equation for the circle \(C\), that is shown in Figure 4. A line through the point \(P ( 8 , - 7 )\) is a tangent to the circle \(C\) at the point \(T\).
2. Find the length of \(P T\).

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.

3. \begin{figure}[h] \end{figure} Diagram not drawn to scale The circle \(C\) has centre \(P ( 7,8 )\) and passes through the point \(Q ( 10,13 )\), as shown in Figure 2.

1. Find the length \(P Q\), giving your answer as an exact value.
2. Hence write down an equation for \(C\). The line \(l\) is a tangent to \(C\) at the point \(Q\), as shown in Figure 2.
3. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y + 30 = 0$$ Find

1. the coordinates of the centre of \(C\),
2. the radius of \(C\),
3. the \(y\) coordinates of the points where the circle \(C\) crosses the line with equation \(x = 4\), giving your answers as simplified surds.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 2 x + 14 y = 0$$ Find

1. the coordinates of the centre of \(C\),
2. the exact value of the radius of \(C\),
3. the \(y\) coordinates of the points where the circle \(C\) crosses the \(y\)-axis.
4. Find an equation of the tangent to \(C\) at the point ( 2,0 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

4. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 12 x\) The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on \(C\), where \(p \neq 0\)

1. Show that the equation of the normal to the curve \(C\) at the point \(P\) is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve \(C\) again at the point \(Q\).
   Given that \(p = 2\) and that \(S\) is the focus of the parabola, find
2. the coordinates of the point \(Q\),
3. the area of the triangle \(P Q S\).

2. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
The hyperbola \(H\) has eccentricity \(\frac { \sqrt { 21 } } { 4 }\) and passes through the point (12, 5).
Find

1. the value of \(a\) and the value of \(b\),
2. the coordinates of the foci of \(H\).

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),

1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).

2. An ellipse has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 2 \sin \theta ) , 0 < \theta < \frac { \pi } { 2 }\) The line \(L\) is a normal to the ellipse at the point \(P\).

1. Show that an equation for \(L\) is $$5 x \sin \theta - 2 y \cos \theta = 21 \sin \theta \cos \theta$$ Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(P Q\),
2. find the exact area of triangle \(O P M\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2 \theta\) WIHN SIHI NITIIUM ION OC
   VIUV SIHI NI JAHM ION OC
   VI4V SIHI NIS IIIM ION OC

A circle, with centre \(C\) and radius \(r\), has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 4 y - 12 = 0$$ Find

1. the coordinates of \(C\),
2. the exact value of \(r\). The circle cuts the \(y\)-axis at the points \(A\) and \(B\).
3. Find the coordinates of the points \(A\) and \(B\).

Solve the simultaneous equations $$\begin{gathered} y - 2 x - 4 = 0 \\ 4 x ^ { 2 } + y ^ { 2 } + 20 x = 0 \end{gathered}$$

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 \(C\).

1. The line \(x = 8\) is a directrix of the ellipse with equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > 0 , b > 0$$ and the point \(( 2,0 )\) is the corresponding focus.
Find the value of \(a\) and the value of \(b\).

1. The hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ Find

1. the coordinates of the foci of \(H\),
2. the equations of the directrices of \(H\).

1. The hyperbola \(H\) has foci at \(( 5,0 )\) and \(( - 5,0 )\) and directrices with equations \(x = \frac { 9 } { 5 }\) and \(x = - \frac { 9 } { 5 }\).

Find a cartesian equation for \(H\).

1. The point \(P\) lies on the ellipse \(E\) with equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$$ \(N\) is the foot of the perpendicular from point \(P\) to the line \(x = 8\) \(M\) is the midpoint of \(P N\).

1. Sketch the graph of the ellipse \(E\), showing also the line \(x = 8\) and a possible position for the line \(P N\).
2. Find an equation of the locus of \(M\) as \(P\) moves around the ellipse.
3. Show that this locus is a circle and state its centre and radius.

1. A hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1 , \quad \text { where } a \text { is a positive constant. }$$ The foci of \(H\) are at the points with coordinates \(( 13,0 )\) and \(( - 13,0 )\).
Find

1. the value of the constant \(a\),
2. the equations of the directrices of \(H\).

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 \(C\).

10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.

1. Calculate the gradient of \(D E\).
2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
4. Calculate the length of \(D F\).
5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).

5

1. Express \(x ^ { 2 } + 3 x\) in the form \(( x + a ) ^ { 2 } + b\).
2. Express \(y ^ { 2 } - 4 y - \frac { 11 } { 4 }\) in the form \(( y + p ) ^ { 2 } + q\). A circle has equation \(x ^ { 2 } + y ^ { 2 } + 3 x - 4 y - \frac { 11 } { 4 } = 0\).
3. Write down the coordinates of the centre of the circle.
4. Find the radius of the circle.