Maximum/minimum distance from pole or line

6 \includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_405_914_260_612} The diagram shows the curve \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 \left( x ^ { 2 } - y ^ { 2 } \right)\) and one of its maximum points \(M\). Find the coordinates of \(M\).

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.

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

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 half-line \(\theta = 2\).
   Now consider the part of \(C\) where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
3. Show that, at the point furthest from the half-line \(\theta = \frac { 1 } { 2 } \pi\), $$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$ and verify that this equation has a root between 0.6 and 0.7 . \(7 \quad\) The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{array} \right)\).
   1. Find the set of values of \(k\) for which \(\mathbf { A }\) is non-singular.
   2. Given that \(\mathbf { A }\) is non-singular, find, in terms of \(k\), the entries in the top row of \(\mathbf { A } ^ { - 1 }\).
   3. Given that \(\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right)\), give an example of a matrix \(\mathbf { C }\) such that \(\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\).
   4. Find the set of values of \(k\) for which the transformation in the \(x - y\) plane represented by \(\left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\) has two distinct invariant lines through the origin.
      If you use the following page to complete the answer to any question, the question number must be clearly shown.

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

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Find the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \pi\).
3. Show that, at the point of \(C\) furthest from the initial line, $$\left( \theta + \frac { 1 } { \theta } \right) \cot \theta - 1 = 0$$ and verify that this equation has a root between 1.1 and 1.2.

11 The curve \(C\) has polar equation $$r = \frac { a } { 1 + \theta }$$ where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(r\) decreases as \(\theta\) increases.
2. The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\), $$\tan \theta = 1 + \theta$$ and verify that this equation has a root between 1.1 and 1.2.
3. Draw a sketch of \(C\).
4. Find the area of the region bounded by the initial line, the line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).

The curve \(C _ { 1 }\) has polar equation \(r ^ { 2 } = 2 \theta\), 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 \(P\), $$2 \theta \tan \theta = 1$$ and verify that this equation has a root between 0.6 and 0.7 .
   The curve \(C _ { 2 }\) has polar equation \(r ^ { 2 } = \theta \sec ^ { 2 } \theta\), for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
2. Find the exact value of \(\theta\) at \(Q\).
3. The diagram below shows the curve \(C _ { 2 }\). Sketch \(C _ { 1 }\) on this diagram.
4. Find, in exact form, the area of the region \(O P Q\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).

10 The curve \(C\) has polar equation $$r = a \sin 3 \theta$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).

1. Show that the area of the region enclosed by \(C\) is \(\frac { 1 } { 12 } \pi a ^ { 2 }\).
2. Show that, at the point of \(C\) at maximum distance from the initial line, $$\tan 3 \theta + 3 \tan \theta = 0 .$$
3. Use the formula $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ to find this maximum distance.
4. Draw a sketch of \(C\).

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 \(\theta = \frac { 1 } { 4 } \pi\).

14 A curve has polar equation \(\mathrm { r } = \mathrm { a } ( \cos \theta + 2 \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \pi\).

1. Determine the polar coordinates of the point on the curve which is furthest from the pole.
2. 1. Show that the curve is a circle whose radius should be specified.
   2. Write down the polar coordinates of the centre of the circle.

9 The curve \(C\) has polar equation $$r = 5 \sqrt { } ( \cot \theta ) ,$$ where \(0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to 1 decimal place.
   Let \(P\) be the point on \(C\) where \(\theta = 0.01\).
2. Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
3. Find the maximum distance of \(C\) from the initial line.
4. Sketch \(C\).

1. Show that the curve with Cartesian equation $$\left(x^2+y^2\right)^2 = 6xy$$ has polar equation \(r^2 = 3\sin 2\theta\). [2]

The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
2. Find the area of the region enclosed by \(C\). [2]
3. Find the maximum distance of a point on \(C\) from the initial line. [6]

1. Show that the curve with Cartesian equation \(\left(x^2 + y^2\right)^2 = 6xy\) has polar equation \(r^2 = 3\sin 2\theta\). [2]

The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
2. Find the area of the region enclosed by \(C\). [2]
3. Find the maximum distance of a point on \(C\) from the initial line. [6]