1.03d Circles: equation (x-a)^2+(y-b)^2=r^2

7 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0\).

1. Find:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle in the form \(p \sqrt { 2 }\), where \(p\) is an integer.
2. A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
3. A line has equation \(y = 2 k - x\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
   2. Find the values of \(k\) for which the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (c)(ii).

5 A circle with centre \(C ( - 5,6 )\) touches the \(y\)-axis, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-6_444_698_372_680}

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. 1. Verify that the point \(P ( - 2,2 )\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(P\).
   3. The mid-point of \(P C\) is \(M\). Determine whether the point \(P\) is closer to the point \(M\) or to the origin \(O\).

8 A circle has centre \(C ( 3 , - 8 )\) and radius 10 .

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis.
3. The tangent to the circle at the point \(A\) has gradient \(\frac { 5 } { 2 }\). Find an equation of the line \(C A\), giving your answer in the form \(r x + s y + t = 0\), where \(r , s\) and \(t\) are integers.
4. The line with equation \(y = 2 x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.

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. \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.

The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).

1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
2. Find the \(x\)-coordinates of \(A\) and \(B\).
3. Show that \(A B = 2 \sqrt { 10 }\).
   

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.

8. \begin{figure}[h] \end{figure} Figure 2 shows the circle \(C\) and the straight line \(l\). The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).

1. Find the gradient of \(l\).
2. Find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the centre of \(C\).
4. Show that \(C\) has the equation \(x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0\).

6. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0 .$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\).
3. Sketch C. Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).

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.

1. The point \(A\) has coordinates ( 4,6 ).

Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),

1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
2. Find the coordinates of \(B\).
3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   

8 A curve \(C\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$

1. Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
2. Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
3. Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
4. 1. Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$16 x ^ { 2 } - 200 x + 625 = 0$$
   2. Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).

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}

9 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows the curve with equation $$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$ and the straight line with equation $$x + y = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}

1. Write down the exact coordinates of the points where the curve with equation \(\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1\) intersects the coordinate axes.
2. The curve is translated \(k\) units in the positive \(x\) direction, where \(k\) is a constant. Write down, in terms of \(k\), the equation of the curve after this translation.
3. Show that, if the line \(x + y = 2\) intersects the translated curve, the \(x\)-coordinates of the points of intersection must satisfy the equation $$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
4. Hence find the two values of \(k\) for which the line \(x + y = 2\) is a tangent to the translated curve. Give your answer in the form \(p \pm \sqrt { q }\), where \(p\) and \(q\) are integers.
5. On Figure 3, show the translated curves corresponding to these two values of \(k\). \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2 (for use in Question 5)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3 (for use in Question 9)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
   \end{figure}

6 An ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Sketch the ellipse \(E\), showing the coordinates of the points of intersection of the ellipse with the coordinate axes.
2. The ellipse \(E\) is stretched with scale factor 2 parallel to the \(y\)-axis. Find and simplify the equation of the curve after the stretch.
3. The original ellipse, \(E\), is translated by the vector \(\left[ \begin{array} { l } a \\ b \end{array} \right]\). The equation of the translated ellipse is $$4 x ^ { 2 } + 3 y ^ { 2 } - 8 x + 6 y = 5$$ Find the values of \(a\) and \(b\).

1

1. A family of curves is given by the equation $$x ^ { 2 } + y ^ { 2 } + 2 a x y = 1 ( * )$$ where the parameter \(a\) is a real number.
   You may find it helpful to use a slider (for \(a\) ) to investigate this family of curves.
   1. On the axes in the Printed Answer Booklet, sketch the curve in each of the cases
      • \(a = 0\)
      • \(a = 0.5\)
      • \(a = 2\)
      • State a feature of the curve for the cases \(a = 0 , a = 0.5\) that is not a feature of the curve in the case \(a = 2\).
      • In the case \(a = 1\), the curve consists of two straight lines. Determine the equations of these lines.
      • 1. Find an equation of the curve (*) in polar form.
        2. Hence, or otherwise, find, in exact form, the area bounded by the curve, the positive part of the \(x\)-axis and the positive part of the \(y\)-axis, in the case \(a = 2\).
2. In this part of the question \(m\) is any real number.
Describing all possible cases, determine the pairs of values \(a\) and \(m\) for which the curve with equation (*) intersects the straight line given by \(y = m x\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\)

The point \(P\) on \(C\) has \(y\) coordinate \(p\), where \(p\) is a positive constant.

1. Show that an equation of the tangent to \(C\) at \(P\) is given by $$2 p y = 16 x + p ^ { 2 }$$ $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
2. Write down the equation of the directrix of \(C\). The line \(l\) is the reflection of the tangent to \(C\) at \(P\) in the directrix of \(C\).
   Given that \(l\) passes through the focus of \(C\),
3. determine the exact value of \(p\).

7. \begin{figure}[h] \end{figure} A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student's design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = 1\), the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) through \(360 ^ { \circ }\) about the \(y\)-axis. The point \(A\) has coordinates ( \(1,0.5\) ) and the point \(B\) has coordinates ( \(0.5,2.5\) ) where the units are centimetres. The curve \(C _ { 1 }\) is modelled by the equation $$x = \frac { a } { y + b } \quad 0.5 \leqslant y \leqslant 2.5$$

1. Determine the value of \(a\) and the value of \(b\) according to the model. The curve \(C _ { 2 }\) is modelled to be an arc of the circle with centre \(( 0,3 )\).
2. Use calculus to determine the volume of plastic required to make the chess piece according to the model.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 16 } = 1$$ The points \(S\) and \(S ^ { \prime }\) are the foci of \(E\).

1. Find the coordinates of \(S\) and \(S ^ { \prime }\)
2. Show that for any point \(P\) on \(E\), the triangle \(P S S ^ { \prime }\) has constant perimeter and determine its value.

1. The points \(P \left( 9 p ^ { 2 } , 18 p \right)\) and \(Q \left( 9 q ^ { 2 } , 18 q \right) , p \neq q\), lie on the parabola \(C\) with equation

$$y ^ { 2 } = 36 x$$ The line \(l\) passes through the points \(P\) and \(Q\)

1. Show that an equation for the line \(l\) is $$( p + q ) y = 2 ( x + 9 p q )$$ The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(A\).
2. Show that the coordinates of \(A\) are $$\left( 9 \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - 9 p q ( p + q ) \right)$$ Given that the points \(P\) and \(Q\) vary such that \(l\) always passes through the point \(( 12,0 )\)
3. find, in the form \(y ^ { 2 } = \mathrm { f } ( x )\), an equation for the locus of \(A\), giving \(\mathrm { f } ( x )\) in simplest form.