Conic translation and transformation

5 The curve with equation \(y = \frac { x ^ { 2 } - k x + 2 k } { x + k }\) is to be investigated for different values of \(k\).

1. Use your graphical calculator to obtain rough sketches of the curve in the cases \(k = - 2\), \(k = - 0.5\) and \(k = 1\).
2. Show that the equation of the curve may be written as \(y = x - 2 k + \frac { 2 k ( k + 1 ) } { x + k }\). Hence find the two values of \(k\) for which the curve is a straight line.
3. When the curve is not a straight line, it is a conic.
   (A) Name the type of conic.
   (B) Write down the equations of the asymptotes.
4. Draw a sketch to show the shape of the curve when \(1 < k < 8\). This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where \(x = 1\) and \(x = k\) respectively.

5 A curve has parametric equations \(x = \lambda \cos \theta - \frac { 1 } { \lambda } \sin \theta , y = \cos \theta + \sin \theta\), where \(\lambda\) is a positive constant.

1. Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = 0.5 , \quad \lambda = 3 \quad \text { and } \quad \lambda = 5 .$$
2. Given that the curve is a conic, name the type of conic.
3. Show that \(y\) has a maximum value of \(\sqrt { 2 }\) when \(\theta = \frac { 1 } { 4 } \pi\).
4. Show that \(x ^ { 2 } + y ^ { 2 } = \left( 1 + \lambda ^ { 2 } \right) + \left( \frac { 1 } { \lambda ^ { 2 } } - \lambda ^ { 2 } \right) \sin ^ { 2 } \theta\), and deduce that the distance from the origin of any point on the curve is between \(\sqrt { 1 + \frac { 1 } { \lambda ^ { 2 } } }\) and \(\sqrt { 1 + \lambda ^ { 2 } }\).
5. For the case \(\lambda = 1\), show that the curve is a circle, and find its radius.
6. For the case \(\lambda = 2\), draw a sketch of the curve, and label the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H }\) on the curve corresponding to \(\theta = 0 , \frac { 1 } { 4 } \pi , \frac { 1 } { 2 } \pi , \frac { 3 } { 4 } \pi , \pi , \frac { 5 } { 4 } \pi , \frac { 3 } { 2 } \pi , \frac { 7 } { 4 } \pi\) respectively. You should make clear what is special about each of these points. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of \(k\).

1. Firstly consider cases in which \(k\) is a positive even integer.
   (A) State the shape of the curve when \(k = 2\).
   (B) Sketch, on the same axes, the curves for \(k = 2\) and \(k = 4\).
   (C) Describe the shape that the curve tends to as \(k\) becomes very large.
   (D) State the range of possible values of \(x\) and \(y\).
2. Now consider cases in which \(k\) is a positive odd integer.
   (A) Explain why \(x\) and \(y\) may take any value.
   (B) State the shape of the curve when \(k = 1\).
   (C) Sketch the curve for \(k = 3\). State the equation of the asymptote of this curve.
   (D) Sketch the shape that the curve tends to as \(k\) becomes very large.
3. Now let \(k = \frac { 1 } { 2 }\). Sketch the curve, indicating the range of possible values of \(x\) and \(y\).
4. Now consider the modified equation \(| x | ^ { k } + | y | ^ { k } = 1\).
   (A) Sketch the curve for \(k = \frac { 1 } { 2 }\).
   (B) Investigate the shape of the curve for \(k = \frac { 1 } { n }\) as the positive integer \(n\) becomes very large.

6
\includegraphics[max width=\textwidth, alt={}, center]{268b605f-eb86-40df-946a-210da1355e83-3_716_1431_852_356} The diagram shows the curve with equation \(y = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } }\), where \(a\) is a positive constant.

1. Find the equations of the asymptotes of the curve.
2. Sketch the curve with equation $$y ^ { 2 } = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } } .$$ State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.

2
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-2_728_951_486_534} The diagram shows the graph of $$y = \frac { 2 x ^ { 2 } + 3 x + 3 } { x + 1 }$$

1. Find the equations of the asymptotes of the curve.
2. Prove that the values of \(y\) between which there are no points on the curve are - 5 and 3 .

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the parabola with equation \(y = \frac { 1 } { 2 } x ( 10 - x ) , 0 \leqslant x \leqslant 10\) This question concerns rectangles that lie under the parabola in the first quadrant．The bottom edge of each rectangle lies along the \(x\)－axis and the top left vertex lies on the parabola．Some examples are shown in Figure 2. Let the \(x\) coordinate of the top left vertex be \(a\) ．
（a）Explain why the width，\(w\) ，of such a rectangle must satisfy \(w \leqslant 10 - 2 a\)
（b）Find the value of \(a\) that gives the maximum area for such a rectangle． Given that the rectangle must be a square，
（c）find the value of \(a\) that gives the maximum area for such a square． Given that the area of the rectangles is fixed as 36
（d）find the range of possible values for \(a\) ．
\includegraphics[max width=\textwidth, alt={}, center]{4d5b914c-28b2-4485-a42e-627c95fa16e2-16_2255_50_311_1980}

6．（a）Given that \(x ^ { 4 } + y ^ { 4 } = 1\) ，prove that \(x ^ { 2 } + y ^ { 2 }\) is a maximum when \(x = \pm y\) ，and find the maximum and minimum values of \(x ^ { 2 } + y ^ { 2 }\) ．
（b）On the same diagram，sketch the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations \(x ^ { 4 } + y ^ { 4 } = 1\) and \(x ^ { 2 } + y ^ { 2 } = 1\) respectively．
（c）Write down the equation of the circle \(C _ { 3 }\) ，centre the origin，which touches the curve \(C _ { 1 }\) at the points where \(x = \pm y\) ．

4
\includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-2_947_1305_986_420} The diagram shows the curve with equation $$y = \frac { a x + b } { x + c }$$ where \(a , b\) and \(c\) are constants.

1. Given that the asymptotes of the curve are \(x = - 1\) and \(y = - 2\) and that the curve passes through \(( 3,0 )\), find the values of \(a , b\) and \(c\).
2. Sketch the curve with equation $$y ^ { 2 } = \frac { a x + b } { x + c }$$ for the values of \(a , b\) and \(c\) found in part (i). State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.

8 The curve \(C _ { 1 }\) has equation \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are polynomials of degree 2 and 1 respectively. The asymptotes of the curve are \(x = - 2\) and \(y = \frac { 1 } { 2 } x + 1\), and the curve passes through the point \(\left( - 1 , \frac { 17 } { 2 } \right)\).

1. Express the equation of \(C _ { 1 }\) in the form \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\).
2. For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
3. For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
   Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
4. It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of intersection. Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
5. It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of
   intersection. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

7 The 'parabolic' TV satellite dish in the diagram can be modelled by the surface generated by the rotation of part of a parabola around a vertical \(z\)-axis. The model is represented by part of the surface with equation \(z = \mathrm { f } ( x , y )\) and \(O\) is on the surface. The point \(P\) is on the rim of the dish and directly above the \(x\)-axis.
The object, \(B\), modelled as a point on the \(z\)-axis is the receiving box which collects the TV signals reflected by the dish.
\includegraphics[max width=\textwidth, alt={}, center]{f2166e0a-cd4c-40af-b4b4-04ef4919d996-3_753_995_584_525}

1. The horizontal plane \(\Pi _ { 1 }\), containing the point \(P\), intersects the surface of the model in a contour of the surface.
   (a) Sketch this contour in the Printed Answer Booklet.
   (b) State a suitable equation for this contour.
2. A second plane, \(\Pi _ { 2 }\), containing both \(P\) and the \(z\)-axis, intersects the surface of the model in a section of the surface.
   (a) Sketch this section in the Printed Answer Booklet.
   (b) State a suitable equation for this section.
3. A proposed equation for the surface is \(z = a x ^ { 2 } + b y ^ { 2 }\). What can you say about the constants \(a\) and \(b\) within this equation? Justify your answers.
4. The real TV satellite dish has the following measurements (in metres): the height of \(P\) above \(O\) is 0.065 and the perimeter of the rim is 2.652 . Using this information, calculate correct to three decimal places the values of
   • \(a\) and \(b\),
   • any other constants stated within the answers to parts (i)(b) and (ii)(b).
   • Incoming satellite signals arrive at the dish in linear "beams" travelling parallel to the \(z\)-axis. They are then 'bounced' off the dish to the receiving box at \(B\).
   • On the diagram for part (ii)(a) in the Printed Answer Booklet draw some of these beams and mark \(B\).
   • If the values of \(a\) and \(b\) were changed, what would happen?

5 A research student is using 3-D graph-plotting software to model a chain of volcanic islands in the Pacific Ocean. These islands appear above sea-level at regular intervals, (approximately) distributed along a straight line. Each island takes the form of a single peak; also, along the line of islands, the heights of these peaks decrease in size in an (approximately) regular fashion (see Fig. 1.1). \begin{figure}[h] \end{figure} The student's model uses the surface with equation \(\mathrm { z } = \sin \mathrm { x } + \sin \mathrm { y }\), a part of which is shown in Fig. 1.2 below. The surface of the sea is taken to be the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. - Describe two problems with this model.
   • Suggest revisions to this model so that each of these problems is addressed.
   • Still using their original model, the student examines the contour \(z = 2\) for their surface only to find that the software shows what appears to be an empty graph.
   Explain what has happened.

8 A curve has equation \(y ^ { 2 } = 12 x\).

1. Sketch the curve.
2. 1. The curve is translated by 2 units in the positive \(y\) direction. Write down the equation of the curve after this translation.
   2. The original curve is reflected in the line \(y = x\). Write down the equation of the curve after this reflection.
3. 1. Show that if the straight line \(y = x + c\), where \(c\) is a constant, intersects the curve \(y ^ { 2 } = 12 x\), then the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$
   2. Hence find the value of \(c\) for which the straight line is a tangent to the curve.
   3. Using this value of \(c\), find the coordinates of the point where the line touches the curve.
   4. In the case where \(c = 4\), determine whether the line intersects the curve or not.

7 A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 9 } - y ^ { 2 } = 1$$

1. Find the equations of the asymptotes of \(H\).
2. The asymptotes of \(H\) are shown in the diagram opposite. On the same diagram, sketch the hyperbola \(H\). Indicate on your sketch the coordinates of the points of intersection of \(H\) with the coordinate axes.
3. The hyperbola \(H\) is now translated by the vector \(\left[ \begin{array} { r } - 3 \\ 0 \end{array} \right]\).
   1. Write down the equation of the translated curve.
   2. Calculate the coordinates of the two points of intersection of the translated curve with the line \(y = x\).
4. From your answers to part (c)(ii), deduce the coordinates of the points of intersection of the original hyperbola \(H\) with the line \(y = x - 3\).
   \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-4_675_1157_1932_495}

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 \(b\).

6 A parabola with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant, is translated by the vector \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right]\) to give the curve \(C\). The curve \(C\) passes through the point (4, 7).

1. Show that \(a = 2\).
2. Find the values of \(k\) for which the line \(k y = x\) does not meet the curve \(C\).
   [0pt] [6 marks]

2 A curve has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Sketch the curve, showing the coordinates of the points of intersection with the coordinate axes.
2. Calculate the \(y\)-coordinates of the points of intersection of the curve with the line \(x = 1\). Give your answers in the form \(p \sqrt { 2 }\), where \(p\) is a rational number.
3. The curve is translated one unit in the positive \(x\) direction. Write down the equation of the curve after the translation.

5 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$$ and its asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-03_531_1013_616_516}

1. Write down the equations of the two asymptotes.
2. The points on the hyperbola for which \(x = 4\) are denoted by \(A\) and \(B\). Find, in surd form, the \(y\)-coordinates of \(A\) and \(B\).
3. The hyperbola and its asymptotes are translated by two units in the positive \(y\) direction. Write down:
   1. the \(y\)-coordinates of the image points of \(A\) and \(B\) under this translation;
   2. the equations of the hyperbola and the asymptotes after the translation.

7

1. Describe a geometrical transformation by which the hyperbola $$x ^ { 2 } - 4 y ^ { 2 } = 1$$ can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).
2. The diagram shows the hyperbola \(H\) with equation $$x ^ { 2 } - y ^ { 2 } - 4 x + 3 = 0$$
   \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-4_951_1216_824_402}
   By completing the square, describe a geometrical transformation by which the hyperbola \(H\) can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).

5

1. Show that the equation of \(H _ { 1 }\) can be written in the form $$( x - 1 ) ^ { 2 } - \frac { y ^ { 2 } } { q } = r$$ where \(q\) and \(r\) are integers.
   5
2. \(\quad \mathrm { H } _ { 2 }\) is the hyperbola $$x ^ { 2 } - y ^ { 2 } = 4$$ Describe fully a sequence of two transformations which maps the graph of \(H _ { 2 }\) onto the graph of \(H _ { 1 }\)
   [0pt] [4 marks]

14 The curve \(C\) has polar equation $$r = \frac { 4 } { 5 + 3 \cos \theta } \quad ( - \pi < \theta \leq \pi )$$ 14

1. Show that \(r\) takes values in the range \(\frac { 1 } { k } \leq r \leq k\), where \(k\) is an integer.
   [0pt] [2 marks] 14
2. Find the Cartesian equation of \(C\) in the form \(y ^ { 2 } = \mathrm { f } ( x )\) 14
3. The ellipse \(E\) has equation $$y ^ { 2 } + \frac { 16 x ^ { 2 } } { 25 } = 1$$ Find the transformation that maps the graph of \(E\) onto \(C\)
   [0pt] [4 marks]
   

   15	Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 4 y = \cos 2 x + 5 x\)

6 The ellipse \(E _ { 1 }\) has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$ \(E _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the ellipse \(E _ { 2 }\)
6

1. Write down the equation of \(E _ { 2 }\) 6
2. The ellipse \(E _ { 3 }\) has equation $$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$ Describe the transformation that maps \(E _ { 2 }\) to \(E _ { 3 }\) 6
3. Each of the lines \(L _ { A }\) and \(L _ { B }\) is a tangent to both \(E _ { 2 }\) and \(E _ { 3 }\)
   \(L _ { A }\) is closer to the origin than \(L _ { B }\)
   \(E _ { 2 }\) and \(E _ { 3 }\) both lie between \(L _ { A }\) and \(L _ { B }\)
   Sketch and label \(E _ { 2 } , E _ { 3 } , L _ { A }\) and \(L _ { B }\) on the axes below.
   You do not need to show the values of the axis intercepts for \(L _ { A }\) and \(L _ { B }\)
   \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475} 6
4. Explain, without doing any calculations, why \(L _ { A }\) has an equation of the form $$x + y = c$$ where \(c\) is a constant.

10 The curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$$ The curve \(C _ { 2 }\) has equation $$x ^ { 2 } - 25 y ^ { 2 } - 6 x - 200 y - 416 = 0$$ 10

1. Find a sequence of transformations that maps the graph of \(C _ { 1 }\) onto the graph of \(C _ { 2 }\) [4 marks]
   10
2. Find the equations of the asymptotes to \(C _ { 2 }\)
   Give your answers in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

5 Josh and Zoe are solving the following mathematics problem: The curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The matrix \(\mathbf { M } = \left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]\) maps \(C _ { 1 }\) onto \(C _ { 2 }\)
Find the equations of the asymptotes of \(C _ { 2 }\) Josh says that to solve this problem you must first carry out the transformation on \(C _ { 1 }\) to find \(C _ { 2 }\), and then find the asymptotes of \(C _ { 2 }\) Zoe says that you will get the same answer if you first find the asymptotes of \(C _ { 1 }\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C _ { 2 }\) Show that Zoe is correct.