4.09a Polar coordinates: convert to/from cartesian

7 The curve \(C _ { 1 }\) has polar equation \(r = \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta - 1 = 0$$ and verify that this equation has a root between 0.6 and 0.7 .
   The curve \(C _ { 2 }\) has polar equation \(r = \theta \sin \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
2. Find the polar coordinates of \(Q\), giving your answers in exact form.
3. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
4. Find, in terms of \(\pi\), the area of the region bounded by the arc \(O Q\) of \(C _ { 1 }\) and the arc \(O Q\) of \(C _ { 2 }\). [7]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The curve \(C\) has polar equation \(r = \operatorname { atan } \theta\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

1. Sketch \(C\) and state the greatest distance of a point on \(C\) from the pole.
2. Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
3. Show that \(C\) has Cartesian equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } } }\).
4. Using your answer to part (b), deduce the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } a \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { a ^ { 2 } - x ^ { 2 } } } d x\).

5 The curve \(C\) has polar equation \(r = \operatorname { acot } \left( \frac { 1 } { 3 } \pi - \theta \right)\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi\). It is given that the greatest distance of a point on \(C\) from the pole is \(2 \sqrt { 3 }\).

1. Sketch \(C\) and show that \(a = 2\).
2. Find the exact value of the area of the region bounded by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 6 } \pi\).
3. Show that \(C\) has Cartesian equation \(2 ( x + y \sqrt { 3 } ) = ( x \sqrt { 3 } - y ) \sqrt { x ^ { 2 } + y ^ { 2 } }\).

6 The curve \(C\) has Cartesian equation \(x ^ { 2 } + x y + y ^ { 2 } = a\), where \(a\) is a positive constant.

1. Show that the polar equation of \(C\) is \(r ^ { 2 } = \frac { 2 a } { 2 + \sin 2 \theta }\).
2. Sketch the part of \(C\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is enclosed by this part of \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
3. It is given that \(\sin 2 \theta\) may be expressed as \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta }\). Use this result to show that the area of \(R\) is $$\frac { 1 } { 2 } a \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \tan ^ { 2 } \theta } { 1 + \tan \theta + \tan ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution \(t = \tan \theta\) to find the exact value of this area.

5

1. Show that the curve with Cartesian equation $$x ^ { 2 } - y ^ { 2 } = a$$ where \(a\) is a positive constant, has polar equation \(r ^ { 2 } = a \sec 2 \theta\).
   The curve \(C\) has polar equation \(r ^ { 2 } = \operatorname { asec } 2 \theta\), where \(a\) is a positive constant, for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).
2. Sketch \(C\) and state the minimum distance of \(C\) from the pole.
3. Find, in terms of \(a\), the exact value of the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \frac { 1 } { 12 } \pi\). [You may use any result from the list of formulae (MF19) without proof.] [4]

7 The curve \(C\) has polar equation \(r ^ { 2 } = \sin 2 \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the equation of the line of symmetry.
2. Find a Cartesian equation for \(C\).
   
3. Find the total area enclosed by \(C\).
4. Find the greatest distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-16_2718_36_141_2011} If you use the following page to complete the answer to any question, the question number must be clearly shown.
   

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_143_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_714_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_438_29_1283_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_1852_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_2423_2014}

7

1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 5 } { 2 } } = 4 x y \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r = \sin 4 \theta\).
   The curve \(C\) has polar equation \(r = \sin 4 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
2. Sketch \(C\) and state the equation of the line of symmetry.
3. Find the exact value of the area of the region enclosed by \(C\).
4. Using the identity \(\sin 4 \theta \equiv 4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta\), find the maximum distance of \(C\) from the line \(\theta = \frac { 1 } { 2 } \pi\). Give your answer correct to 2 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The curve \(C\) has polar equation \(r = \ln ( 1 + \pi - \theta )\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Using the substitution \(u = 1 + \pi - \theta\), or otherwise, show that the area of the region enclosed by \(C\) and the initial line is $$\frac { 1 } { 2 } ( 1 + \pi ) \ln ( 1 + \pi ) ( \ln ( 1 + \pi ) - 2 ) + \pi$$
3. Show that, at the point of \(C\) furthest from the initial line, $$( 1 + \pi - \theta ) \ln ( 1 + \pi - \theta ) - \tan \theta = 0$$ and verify that this equation has a root between 1.2 and 1.3.

6 The curve \(C\) has polar equation \(r = 2 \cos \theta ( 1 + \sin \theta )\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Find the polar coordinates of the point on \(C\) that is furthest from the pole.
2. Sketch C.
3. Find the area of the region bounded by \(C\) and the initial line, giving your answer in exact form.

5 The curve \(C\) has polar equation \(r = 3 + 2 \sin \theta\), for \(- \pi < \theta \leqslant \pi\).

1. The diagram shows part of \(C\). Sketch the rest of \(C\) on the diagram.
   
   [diagram]
   The straight line \(l\) has polar equation \(r \sin \theta = 2\).
2. Add \(l\) to the diagram in part (a) and find the polar coordinates of the points of intersection of \(C\) and \(l\).
3. The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\), giving your answer in exact form.

6

1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 36 \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\).
   The curve \(C\) has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
2. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole.
3. Find the area of the region enclosed by \(C\).
4. Find the maximum distance of a point on \(C\) from the initial line, giving the answer in exact form.

5 The curve \(C\) has polar equation \(r = \operatorname { asec } ^ { 2 } \theta\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

1. Sketch \(C\), stating the polar coordinates of the point of intersection of \(C\) with the initial line and also with the half-line \(\theta = \frac { 1 } { 4 } \pi\).
2. Find the maximum distance of a point of \(C\) from the initial line.
3. Find the area of the region enclosed by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
4. Find, in the form \(y = f ( x )\), the Cartesian equation of \(C\).

6

1. Show that the curve with Cartesian equation $$\left( x - \frac { 1 } { 2 } \right) ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 }$$ has polar equation \(r = \cos \theta\).
   The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \cos \theta \quad \text { and } \quad r = \sin 2 \theta$$ respectively, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole and at another point \(P\).
2. Find the polar coordinates of \(P\).
3. In a single diagram sketch \(C _ { 1 }\) and \(C _ { 2 }\), clearly identifying each curve, and mark the point \(P\).
4. The region \(R\) is enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) and includes the line \(O P\). Find, in exact form, the area of \(R\).

6 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { - \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
2. Find the exact value of the area of the region bounded by \(C\) and the initial line.
3. Show that, at the point on \(C\) furthest from the initial line, $$1 - e ^ { \theta - \frac { 1 } { 2 } \pi } - \tan \theta = 0$$ and verify that this equation has a root between 0.56 and 0.57 .

7 The curve \(C _ { 1 }\) has polar equation \(r = a ( \cos \theta + \sin \theta )\) for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.

1. Find a Cartesian equation for \(C _ { 1 }\) and show that it represents a circle, stating its radius and the Cartesian coordinates of its centre.
2. Sketch \(C _ { 1 }\) and state the greatest distance of a point on \(C _ { 1 }\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-14_2721_40_107_2010} The curve \(C _ { 2 }\) with polar equation \(r = a \theta\) intersects \(C _ { 1 }\) at the pole and the point with polar coordinates \(( a \phi , \phi )\).
3. Verify that \(1.25 < \phi < 1.26\).
4. Show that the area of the smaller region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is equal to $$\frac { 1 } { 2 } a ^ { 2 } \left( \frac { 3 } { 4 } \pi + \frac { 1 } { 3 } \phi ^ { 3 } - \phi + \frac { 1 } { 2 } \cos 2 \phi \right)$$ and deduce, in terms of \(a\) and \(\phi\), the area of the larger region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\) and the initial line.
3. Show that the Cartesian equation of \(C\) can be expressed as \(4 \left( x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }\).

8. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 1 - \sin \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\), such that the tangent to \(C\) at \(P\) is parallel to the initial line.

1. Use calculus to determine the polar coordinates of \(P\) The finite region \(R\), shown shaded in Figure 1, is bounded by
   $$\frac { 1 } { 32 } ( a \pi + b \sqrt { 3 } + c )$$ where \(a\), \(b\) and \(c\) are integers to be determined.

9. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C _ { 1 }\) with polar equation \(r = 2 a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and the circle \(C _ { 2 }\) with polar equation \(r = a , 0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant.

1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C _ { 1 }\) meets the circle \(C _ { 2 }\) The regions enclosed by the curve \(C _ { 1 }\) and the circle \(C _ { 2 }\) overlap and the common region \(R\) is shaded in Figure 1.
2. Find the area of the shaded region \(R\), giving your answer in the form \(\frac { 1 } { 12 } a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are integers to be found.

1. The curve \(C\), with pole \(O\), has polar equation

$$r = 1 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the tangent to \(C\) is parallel to the initial line.

1. Find the polar coordinates of \(A\).
2. Find the finite area enclosed by the initial line, the line \(O A\) and the curve \(C\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are rational constants to be found.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ Given that \(C\) meets the initial line at the point \(A\), as shown in Figure 1,

1. write down the polar coordinates of \(A\). The line \(l _ { 1 }\), also shown in Figure 1, is the tangent to \(C\) at the point \(B\) and is parallel to the initial line.
2. Use calculus to determine the polar coordinates of \(B\). The line \(l _ { 2 }\), also shown in Figure 1, is the tangent to \(C\) at \(A\) and is perpendicular to the initial line. The region \(R\), shown shaded in Figure 1, is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\)
3. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(p \sqrt { 3 } + q \pi\) where \(p\) and \(q\) are constants to be determined.

10. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 1 + \cos \theta \quad 0 \leqslant \theta \leqslant \pi$$ and the line \(l\) with polar equation $$r = k \sec \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ where \(k\) is a positive constant.
Given that

• \(\quad C\) and \(l\) intersect at the point \(P\)
• \(O P = 1 + \frac { \sqrt { 3 } } { 2 }\)
  1. determine the exact value of \(k\).

The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the initial line and \(l\).

Use algebraic integration to show that the area of \(R\) is $$p \pi + q \sqrt { 3 } + r$$ where \(p , q\) and \(r\) are simplified rational numbers to be determined.

4. A curve \(C\) has polar equation \(r ^ { 2 } = a ^ { 2 } \cos 2 \theta , 0 \leq \theta \leq \frac { \pi } { 4 }\). The line \(l\) is parallel to the initial line, and \(l\) is the tangent to \(C\) at
above. above.

1. 1. Show that, for any point on \(C , r ^ { 2 } \sin ^ { 2 } \theta\) can be expressed in terms of \(\sin \theta\) and \(a\) only. (1)
   2. Hence, using differentiation, show that the polar coordinates of \(P\) are \(\left( \frac { a } { \sqrt { 2 } } , \frac { \pi } { 6 } \right)\).(6) \includegraphics[max width=\textwidth, alt={}, center]{2352f367-ddf9-4770-ace5-b561b0fbabbb-1_298_725_2163_1169} The shaded region \(R\), shown in the figure above, is bounded by \(C\), the line \(l\) and the half-line with equation \(\theta = \frac { \pi } { 2 }\).
2. Show that the area of \(R\) is \(\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )\).

4. The curve \(C\) has polar equation \(r = 3 a \cos \theta , - \frac { \pi } { 2 } \leq \frac { \pi } { 2 }\). The curve \(D\) has polar equation \(r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). Given that \(a\) is a positive constant, (a) sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
(b) Find the polar coordinates of \(P\) and \(Q\).
(c) Use integration to find the exact area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) The region \(R\) contains all points which lie outside \(D\) and inside \(C\).
Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac { \pi } { 3 }\) is $$\frac { 3 a ^ { 2 } } { 16 } ( 2 \pi - 3 \sqrt { } 3 )$$ (d) show that the area of \(R\) is \(\pi a ^ { 2 }\).

8. \section*{Figure 1} The curve \(C\) shown in Fig. 1 has polar equation $$r = a ( 3 + \sqrt { 5 } \cos \theta ) , \quad - \pi \leq \theta < \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{6d92bf8a-df0d-421c-8246-8160f5921ee6-2_460_792_1503_970}

1. Find the polar coordinates of the points \(P\) and \(Q\) where the tangents to \(C\) are parallel to the initial line. (6) The curve \(C\) represents the perimeter of the surface of a swimming pool. The direct distance from \(P\) to \(Q\) is 20 m.
2. Calculate the value of \(a\).
3. Find the area of the surface of the pool. (6)

7. \begin{figure}[h] \end{figure} A logo is designed which consists of two overlapping closed curves. The polar equations of these curves are \(r = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \cos \boldsymbol { \theta } )\) and $$r = a ( 5 - 2 \cos \theta ) , \quad 0 \leq \theta < 2 \pi .$$ Figure 1 is a sketch (not to scale) of these two curves.

1. Write down the polar corrdinates of the points \(A\) and \(B\) where the curves meet the initial line.(2)
2. Find the polar coordinates of the points \(\boldsymbol { C }\) and \(\boldsymbol { D }\) where the two curves meet. (4)
3. Show that the area of the overlapping region, which is shaded in the figure, is $$\frac { a ^ { 2 } } { 3 } ( 49 \pi - 48 \sqrt { } 3 )$$