4.09a Polar coordinates: convert to/from cartesian

7 The curve \(C\) has equation $$r = 10 \ln ( 1 + \theta )$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Use the substitution \(w = \ln ( 1 + \theta )\) to show that the area of the sector bounded by the line \(\theta = \frac { 1 } { 2 } \pi\) and the arc of \(C\) joining the origin to the point where \(\theta = \frac { 1 } { 2 } \pi\) is $$50 \left( b ^ { 2 } - 2 b + 2 \right) \mathrm { e } ^ { b } - 100$$ where \(b = \ln \left( 1 + \frac { 1 } { 2 } \pi \right)\).

3 The curve \(C\) has polar equation $$r = \left( \frac { 1 } { 2 } \pi - \theta \right) ^ { 2 } ,$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Find the area of the region bounded by \(C\) and the initial line, leaving your answer in terms of \(\pi\).

8 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations given by $$\begin{array} { l l r } C _ { 1 } : & r = 3 \sin \theta , & 0 \leqslant \theta < \pi , \\ C _ { 2 } : & r = 1 + \sin \theta , & - \pi < \theta \leqslant \pi . \end{array}$$

1. Find the polar coordinates of the points, other than the pole, where \(C _ { 1 }\) and \(C _ { 2 }\) meet.
2. In a single diagram, draw sketch graphs of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Show that the area of the region which is inside \(C _ { 1 }\) but outside \(C _ { 2 }\) is \(\pi\).

1 Find the cartesian equation corresponding to the polar equation \(r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)\). Sketch the the graph of \(r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)\), for \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\), indicating clearly the polar coordinates of the intersection with the initial line.

10 The curve \(C\) has polar equation \(r = 2 \sin \theta ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\). Find \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) and hence find the polar coordinates of the point of \(C\) that is furthest from the pole. Sketch \(C\). Find the exact area of the sector from \(\theta = 0\) to \(\theta = \frac { 1 } { 4 } \pi\).

8 A circle has polar equation \(r = a\), for \(0 \leqslant \theta < 2 \pi\), and a cardioid has polar equation \(r = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant. Draw sketches of the circle and the cardioid on the same diagram. Write down the polar coordinates of the points of intersection of the circle and the cardioid. Show that the area of the region that is both inside the circle and inside the cardioid is $$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$

A curve \(C\) has parametric equations $$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$ Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }\). Hence find

1. the arc length of \(C\),
2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use the fact that \(t = \frac { y } { x }\) to find a cartesian equation of \(C\). Hence show that the polar equation of \(C\) is \(r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)\), and state the domain of \(\theta\). Find the area of the region enclosed between \(C\) and the initial line. {www.cie.org.uk} after the live examination series. }

The polar equation of a curve \(C\) is \(r = a ( 1 + \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Show that the cartesian equation of \(C\) is $$x ^ { 2 } + y ^ { 2 } = a \left( x + \sqrt { } \left( x ^ { 2 } + y ^ { 2 } \right) \right)$$
3. Find the area of the sector of \(C\) between \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
4. Find the arc length of \(C\) between the point where \(\theta = 0\) and the point where \(\theta = \frac { 1 } { 3 } \pi\).

The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = 2 \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right) , \\ & C _ { 2 } : r = \mathrm { e } ^ { 2 \theta } - \mathrm { e } ^ { - 2 \theta } \end{aligned}$$ The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(\mathrm { e } ^ { 2 \alpha } - 2 \mathrm { e } ^ { \alpha } - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4 \sqrt { } 2\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the initial line, giving your answer correct to 3 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\).
3. Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\).

6 A curve \(C\) has polar equation $$r ^ { 2 } \sin 2 \theta = 8$$

1. Find the cartesian equation of \(C\) in the form \(y = \mathrm { f } ( x )\).
2. Sketch the curve \(C\).
3. The line with polar equation \(r = 2 \sec \theta\) intersects \(C\) at the point \(A\). Find the polar coordinates of \(A\).

5 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_301_668_1644_689} The curve \(C\) has polar equation $$r = \frac { 2 } { 3 + 2 \cos \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$

1. Verify that the point \(L\) with polar coordinates ( \(2 , \pi\) ) lies on \(C\).
2. The circle with polar equation \(r = 1\) intersects \(C\) at the points \(M\) and \(N\).
   1. Find the polar coordinates of \(M\) and \(N\).
   2. Find the area of triangle \(L M N\).
3. Find a cartesian equation of \(C\), giving your answer in the form \(9 y ^ { 2 } = \mathrm { f } ( x )\).

8 The diagram shows a sketch of a curve \(C\) and a line \(L\), which is parallel to the initial line and touches the curve at the points \(P\) and \(Q\). \includegraphics[max width=\textwidth, alt={}, center]{32de7ef6-b7aa-4bfd-a73a-e12bfc0147e2-5_506_762_447_639} The polar equation of the curve \(C\) is $$r = 4 ( 1 - \sin \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ and the polar equation of the line \(L\) is $$r \sin \theta = 1$$

1. Show that the polar coordinates of \(P\) are \(\left( 2 , \frac { \pi } { 6 } \right)\) and find the polar coordinates of \(Q\).
2. Find the area of the shaded region \(R\) bounded by the line \(L\) and the curve \(C\). Give your answer in the form \(m \sqrt { 3 } + n \pi\), where \(m\) and \(n\) are integers.

3 A curve \(C\) has polar equation \(r ( 1 + \cos \theta ) = 2\).

1. Find the cartesian equation of \(C\), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
2. The straight line with polar equation \(4 r = 3 \sec \theta\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find the length of \(P Q\).

8 The diagram shows a sketch of a curve. \includegraphics[max width=\textwidth, alt={}, center]{f05737eb-adb1-4228-aebf-6b5c7f26a434-5_464_574_402_726} The polar equation of the curve is $$r = \sin 2 \theta \sqrt { \left( 2 + \frac { 1 } { 2 } \cos \theta \right) } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) is the point of the curve at which \(\theta = \frac { \pi } { 3 }\). The perpendicular from \(P\) to the initial line meets the initial line at the point \(N\).

1. 1. Find the exact value of \(r\) when \(\theta = \frac { \pi } { 3 }\).
   2. Show that the polar equation of the line \(P N\) is \(r = \frac { 3 \sqrt { 3 } } { 8 } \sec \theta\).
   3. Find the area of triangle \(O N P\) in the form \(\frac { k \sqrt { 3 } } { 128 }\), where \(k\) is an integer.
2. 1. Using the substitution \(u = \sin \theta\), or otherwise, find \(\int \sin ^ { n } \theta \cos \theta \mathrm {~d} \theta\), where \(n \geqslant 2\).
   2. Find the area of the shaded region bounded by the line \(O P\) and the arc \(O P\) of the curve. Give your answer in the form \(a \pi + b \sqrt { 3 } + c\), where \(a , b\) and \(c\) are constants.
      (8 marks)

4 The diagram shows the curve \(C\) with polar equation $$r = 6 ( 1 - \cos \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{06ae13de-5cf3-421d-ac7a-ee9f74b653be-3_552_903_922_550}

1. Find the area of the region bounded by the curve \(C\).
2. The circle with cartesian equation \(x ^ { 2 } + y ^ { 2 } = 9\) intersects the curve \(C\) at the points \(A\) and \(B\).
   1. Find the polar coordinates of \(A\) and \(B\).
   2. Find, in surd form, the length of \(A B\).

3

1. Show that \(x ^ { 2 } = 1 - 2 y\) can be written in the form \(x ^ { 2 } + y ^ { 2 } = ( 1 - y ) ^ { 2 }\).
2. A curve has cartesian equation \(x ^ { 2 } = 1 - 2 y\). Find its polar equation in the form \(r = \mathrm { f } ( \theta )\), given that \(r > 0\).

8 The polar equation of a curve \(C\) is $$r = 5 + 2 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$

1. Verify that the points \(A\) and \(B\), with polar coordinates ( 7,0 ) and ( \(3 , \pi\) ) respectively, lie on the curve \(C\).
2. Sketch the curve \(C\).
3. Find the area of the region bounded by the curve \(C\).
4. The point \(P\) is the point on the curve \(C\) for which \(\theta = \alpha\), where \(0 < \alpha \leqslant \frac { \pi } { 2 }\). The point \(Q\) lies on the curve such that \(P O Q\) is a straight line, where the point \(O\) is the pole. Find, in terms of \(\alpha\), the area of triangle \(O Q B\).

3 The diagram shows a sketch of a circle which passes through the origin \(O\). \includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-3_423_451_356_794} The equation of the circle is \(( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25\) and \(O A\) is a diameter.

1. Find the cartesian coordinates of the point \(A\).
2. Using \(O\) as the pole and the positive \(x\)-axis as the initial line, the polar coordinates of \(A\) are \(( k , \alpha )\).
   1. Find the value of \(k\) and the value of \(\tan \alpha\).
   2. Find the polar equation of the circle \(( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25\), giving your answer in the form \(r = p \cos \theta + q \sin \theta\).

6 The polar equation of a curve \(C _ { 1 }\) is $$r = 2 ( \cos \theta - \sin \theta ) , \quad 0 \leqslant \theta \leqslant 2 \pi$$

1. 1. Find the cartesian equation of \(C _ { 1 }\).
   2. Deduce that \(C _ { 1 }\) is a circle and find its radius and the cartesian coordinates of its centre.
2. The diagram shows the curve \(C _ { 2 }\) with polar equation $$r = 4 + \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{90a59b47-3799-46a2-b76b-ced5cc3e1aac-4_519_847_443_593}
   1. Find the area of the region that is bounded by \(C _ { 2 }\).
   2. Prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
   3. Find the area of the region that is outside \(C _ { 1 }\) but inside \(C _ { 2 }\).

8

1. A curve has cartesian equation \(x y = 8\). Show that the polar equation of the curve is \(r ^ { 2 } = 16 \operatorname { cosec } 2 \theta\).
2. The diagram shows a sketch of the curve, \(C\), whose polar equation is $$r ^ { 2 } = 16 \operatorname { cosec } 2 \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-4_364_567_1635_726}
   1. Find the polar coordinates of the point \(N\) which lies on the curve \(C\) and is closest to the pole \(O\).
   2. The circle whose polar equation is \(r = 4 \sqrt { 2 }\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find, in an exact form, the polar coordinates of \(P\) and \(Q\).
   3. The obtuse angle \(P N Q\) is \(\alpha\) radians. Find the value of \(\alpha\), giving your answer to three significant figures.
      (5 marks)

2 The Cartesian equation of a circle is \(( x + 8 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 100\).
Using the origin \(O\) as the pole and the positive \(x\)-axis as the initial line, find the polar equation of this circle, giving your answer in the form \(r = p \sin \theta + q \cos \theta\).
(4 marks)

8 The diagram shows a sketch of a curve and a circle. \includegraphics[max width=\textwidth, alt={}, center]{a2bc95fe-5588-4ff7-a8a3-0cd07df412c9-4_460_693_370_680} The polar equation of the curve is $$r = 3 + 2 \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ The circle, whose polar equation is \(r = 2\), intersects the curve at the points \(P\) and \(Q\), as shown in the diagram.

1. Find the polar coordinates of \(P\) and the polar coordinates of \(Q\).
2. A straight line, drawn from the point \(P\) through the pole \(O\), intersects the curve again at the point \(A\).
   1. Find the polar coordinates of \(A\).
   2. Find, in surd form, the length of \(A Q\).
   3. Hence, or otherwise, explain why the line \(A Q\) is a tangent to the circle \(r = 2\).
3. Find the area of the shaded region which lies inside the circle \(r = 2\) but outside the curve \(r = 3 + 2 \sin \theta\). Give your answer in the form \(\frac { 1 } { 6 } ( m \sqrt { 3 } + n \pi )\), where \(m\) and \(n\) are integers.

3 A curve has polar equation \(r ( 4 - 3 \cos \theta ) = 4\). Find its Cartesian equation in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
[0pt] [4 marks]

4

1. The curve with Cartesian equation \(\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1\) is mapped onto the curve with polar equation \(r = \frac { 10 } { 3 - 2 \cos \theta }\) by a single geometrical transformation. By writing the polar equation as a Cartesian equation in a suitable form, find the values of the constants \(c\) and \(d\).
2. Hence describe the geometrical transformation referred to in part (a).
   [0pt] [1 mark]