Tangent parallel/perpendicular to initial line

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of curve \(C\) with polar equation $$r = 3 \sin 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) on \(C\) has polar coordinates \(( R , \phi )\). The tangent to \(C\) at \(P\) is perpendicular to the initial line.

1. Show that \(\tan \phi = \frac { 1 } { \sqrt { 2 } }\)
2. Determine the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and the line \(O P\), where \(O\) is the pole.
3. Use calculus to show that the exact area of \(S\) is $$p \arctan \frac { 1 } { \sqrt { 2 } } + q \sqrt { 2 }$$ where \(p\) and \(q\) are constants to be determined. Solutions relying entirely on calculator technology are not acceptable.

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
   • the line with equation \(\theta = \frac { \pi } { 2 }\)
   • the tangent to \(C\) at \(P\)
   • part of the curve \(C\)
   • the initial line
   • Use algebraic integration to show that the area of \(R\) is
   $$\frac { 1 } { 32 } ( a \pi + b \sqrt { 3 } + c )$$ where \(a\), \(b\) and \(c\) are integers to be determined.

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.

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 \(\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )\).

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)

5. (a) Sketch the curve with polar equation \(\quad r = 3 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leq \theta < \frac { \pi } { 4 }\)
(b) Find the area of the smaller finite region enclosed between the curve and the half-line $$\theta = \frac { \pi } { 6 }$$ (c) Find the exact distance between the two tangents which are parallel to the initial line.
(8)(Total 16 marks)

8. (a) Sketch the curve \(C\) with polar equation $$r = 5 + \sqrt { 3 } \cos \theta , \quad 0 \leq \theta \leq 2 \pi$$ (b) Find the polar coordinates of the points where the tangents to \(C\) are parallel to the initial line \(\theta = 0\). Give your answers to 3 significant figures where appropriate.
(c) Using integration, find the area enclosed by the curve \(C\), giving your answer in terms of \(\pi\).

2. The curve \(C\) has polar equation $$r = 1 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line.
Given that \(O\) is the pole, find the exact length of the line \(O P\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a curve \(C\) with polar equation \(r = a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and a half-line \(l\).
The half-line \(l\) meets \(C\) at the pole \(O\) and at the point \(P\). The tangent to \(C\) at \(P\) is parallel to the initial line. The polar coordinates of \(P\) are \(( R , \phi )\).

1. Show that \(\cos \phi = \frac { 1 } { \sqrt { 3 } }\)
2. Find the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and \(l\).
3. Use calculus to show that the exact area of \(S\) is $$\frac { 1 } { 36 } a ^ { 2 } \left( 9 \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) + \sqrt { 2 } \right)$$

4. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) with polar equation $$r = 2 \cos 2 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 4 }$$ The line \(l\) is parallel to the initial line and is a tangent to \(C\). Find an equation of \(l\), giving your answer in the form \(r = \mathrm { f } ( \theta )\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with polar equation $$r = 1 + \tan \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The tangent to the curve \(C\) at the point \(P\) is perpendicular to the initial line.

1. Find the polar coordinates of the point \(P\). The point \(Q\) lies on the curve \(C\), where \(\theta = \frac { \pi } { 3 }\)
   The shaded region \(R\) is bounded by \(O P , O Q\) and the curve \(C\), as shown in Figure 1
2. Find the exact area of \(R\), giving your answer in the form $$\frac { 1 } { 2 } ( \ln p + \sqrt { q } + r )$$ where \(p , q\) and \(r\) are integers to be found.

7. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \sqrt { 3 } \cos \theta , \quad 0 \leqslant \theta < 2 \pi$$ The tangent to \(C\) at the point \(P\) is parallel to the initial line.

1. Show that \(O P = \frac { 1 } { 2 } ( 3 + \sqrt { 7 } )\)
2. Find the exact area enclosed by the curve \(C\).

6. The curve \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta , \quad \frac { - \pi } { 4 } \leq \theta \leq \frac { \pi } { 4 }$$

1. Sketch the curve \(C\).
2. Find the polar coordinates of the points where tangents to \(C\) are parallel to the initial line.
3. Find the area of the region bounded by \(C\).

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

1. Use calculus to determine the polar coordinates of \(P\). The tangent to \(C\) at the point \(Q\) where \(\theta = \frac { \pi } { 2 }\) is parallel to the initial line.
   The tangent to \(C\) at \(Q\) meets the tangent to \(C\) at \(P\) at the point \(S\), as shown in Figure 1.
   The finite region \(R\), shown shaded in Figure 1, is bounded by the line segments \(Q S , S P\) and the curve \(C\).
2. Use algebraic integration to show that the area of \(R\) is $$\frac { 1 } { 32 } ( a \sqrt { 3 } + b \pi )$$ where \(a\) and \(b\) are integers to be determined.
   (6)

5 Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) are set up in the usual way, with the pole at the origin and the initial line along the positive \(x\)-axis, so that \(x = r \cos \theta\) and \(y = r \sin \theta\). A curve has polar equation \(r = k + \cos \theta\), where \(k\) is a constant with \(k \geqslant 1\).

1. Use your graphical calculator to obtain sketches of the curve in the three cases $$k = 1 , k = 1.5 \text { and } k = 4$$
2. Name the special feature which the curve has when \(k = 1\).
3. For each of the three cases, state the number of points on the curve at which the tangent is parallel to the \(y\)-axis.
4. Express \(x\) in terms of \(k\) and \(\theta\), and find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\). Hence find the range of values of \(k\) for which there are just two points on the curve where the tangent is parallel to the \(y\)-axis. The distance between the point ( \(r , \theta\) ) on the curve and the point ( 1,0 ) on the \(x\)-axis is \(d\).
5. Use the cosine rule to express \(d ^ { 2 }\) in terms of \(k\) and \(\theta\), and deduce that \(k ^ { 2 } \leqslant d ^ { 2 } \leqslant k ^ { 2 } + 1\).
6. Hence show that, when \(k\) is large, the shape of the curve is very nearly circular.

8. The curve \(C\) shown in the diagram above has polar equation $$r = 4 ( 1 - \cos \theta ) , 0 \leq \theta \leq \frac { \pi } { 2 }$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the line \(\theta = \frac { \pi } { 2 }\).

1. Show that \(P\) has polar coordinates \(\left( 2 , \frac { \pi } { 3 } \right)\). The curve \(C\) meets the line \(\theta = \frac { \pi } { 2 }\) at the point \(A\). The tangent to \(C\) at the initial line at the point \(N\). The finite region \(R\), shown shaded in
   \includegraphics[max width=\textwidth, alt={}]{863ef52d-ae75-450c-9eab-8102804868f5-2_737_561_1395_1329} the diagram above, is bounded by the initial line, the line \(\theta = \frac { \pi } { 2 }\), the arc \(A P\) of \(C\) and the line \(P N\).
2. Calculate the exact area of \(R\).

The curve \(C\) has cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = a ^ { 2 } \left( x ^ { 2 } - y ^ { 2 } \right)$$ where \(a\) is a positive constant. Show that \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta$$ Sketch \(C\) for \(- \pi < \theta \leqslant \pi\). Find the area of the sector between \(\theta = - \frac { 1 } { 4 } \pi\) and \(\theta = \frac { 1 } { 4 } \pi\). Find the polar coordinates of all points of \(C\) where the tangent is parallel to the initial line.

8. The curve \(C\) has polar equation $$r = \sin 2 \theta , \quad \text { where } \quad 0 < \theta \leqslant \frac { \pi } { 2 }$$

1. Find the polar coordinates of the point on \(C\) at which the tangent is parallel to the initial line. Give your answers correct to three decimal places.
2. Write the coordinates of this point in Cartesian form.

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

1. Sketch the curve \(C\).
2. 1. Show that the values of \(\theta\) at which the tangent to the curve \(r = 2 - \cos \theta\) is parallel to the initial line satisfy the equation $$2 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0$$
   2. Find the polar coordinates of the points where the tangent to the curve \(r = 2 - \cos \theta\) is parallel to the initial line.

1. The curve \(C\) has equation

$$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(a\) and \(p\) are positive constants and \(p > 2\)
There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.

1. Show that the range of possible values for \(p\) is $$2 < p < 4$$
2. Sketch the curve with equation $$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$ John digs a hole in his garden in order to make a pond.
   The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres.
   Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
   Given that the pond is initially empty,
3. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
4. State a limitation of the model.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 1 + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 3 }$$ Figure 1 also shows the tangent to \(C\) at the point \(A\).
This tangent is perpendicular to the initial line.

1. Use differentiation to prove that the polar coordinates of \(A\) are \(\left( 2 , \frac { \pi } { 4 } \right)\) The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the tangent at \(A\) and the initial line.
2. Use calculus to show that the exact area of \(R\) is \(\frac { 1 } { 2 } ( 1 - \ln 2 )\)

1. (a) Sketch the polar curve \(C\), with equation

$$r = 3 + \sqrt { 5 } \cos \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ On your sketch clearly label the pole, the initial line and the value of \(r\) at the point where the curve intersects the initial line. The tangent to \(C\) at the point \(A\), where \(0 < \theta < \frac { \pi } { 2 }\), is parallel to the initial line.
(b) Use calculus to show that at \(A\) $$\cos \theta = \frac { 1 } { \sqrt { 5 } }$$ (c) Hence determine the value of \(r\) at \(A\).

8. \begin{figure}[h] \end{figure} The curve \(C\) shown in Fig. 1 has polar equation $$r = a ( 3 + \sqrt { 5 } \cos \theta ) , \quad - \pi \leq \theta < \pi$$

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)
   [0pt] [P4 June 2002 Qn 8]

15 The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \(( r , \theta )\) where \(0 \leq \theta \leq \frac { \pi } { 2 }\)
The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole.
\(T P Q\) is the tangent to the curve at \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-26_624_730_653_653} 15

1. Show that the gradient of \(T P Q\) is equal to $$\frac { \frac { \mathrm { d } r } { \mathrm {~d} \theta } \sin \theta + r \cos \theta } { \frac { \mathrm {~d} r } { \mathrm {~d} \theta } \cos \theta - r \sin \theta }$$ 15
2. The curve has polar equation $$r = \mathrm { e } ^ { ( \cot b ) \theta }$$ where \(b\) is a constant such that \(0 < b < \frac { \pi } { 2 }\) Use the result of part (a) to show that the angle between the line \(O P\) and the tangent TPQ does not depend on \(\theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-28_2488_1719_219_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin.