Region bounded by curve and tangent lines

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 4 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 } \text { and } \frac { 3 \pi } { 4 } \leqslant \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , Q R , R S\) and \(S P\) are tangents to \(C\), where \(Q R\) and \(S P\) are parallel to the initial line and \(P Q\) and \(R S\) are perpendicular to the initial line.

1. Find the polar coordinates of the points where the tangent SP touches the curve. Give the values of \(\theta\) to 3 significant figures.
2. Find the exact area of the finite region bounded by the curve \(C\), shown unshaded in Figure 1.
3. Find the area enclosed by the rectangle \(P Q R S\) but outside the curve \(C\), shown shaded in Figure 1.

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

1. Determine the polar coordinates of \(A\). The point \(B\) on the curve has polar coordinates \(\quad a ( 2 + \sqrt { 3 } ) , \frac { \pi } { 6 }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the line \(A B\).
2. Use calculus to determine the exact area of the shaded region \(R\). Give your answer in the form $$\frac { a ^ { 2 } } { 4 } ( d \pi - e + f \sqrt { 3 } )$$ where \(d , e\) and \(f\) are integers.

6. \begin{figure}[h] \end{figure} The curve shown in Figure 1 has polar equation $$r = 4 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta < \pi$$ where \(a\) is a positive constant.
The tangent to the curve at the point \(A\) is parallel to the initial line.

1. Show that the polar coordinates of \(A\) are \(\left( 6 a , \frac { \pi } { 3 } \right)\) The point \(B\) lies on the curve such that angle \(A O B = \frac { \pi } { 6 }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the line \(A B\) and the curve.
2. Use calculus to determine the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( n \pi + p \sqrt { 3 } + q )\), where \(n , p\) and \(q\) are integers.

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.

8. \begin{figure}[h] \end{figure} Figure 1 shows a closed curve \(C\) with equation $$r = 3 ( \cos 2 \theta ) ^ { \frac { 1 } { 2 } } , \quad \text { where } - \frac { \pi } { 4 } < \theta \leqslant \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } < \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , S R , P S\) and \(Q R\) are tangents to \(C\), where \(P Q\) and \(S R\) are parallel to the initial line and \(P S\) and \(Q R\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1.
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1.

\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3(\cos 2\theta)^{\frac{1}{2}}, \quad \text{where } -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}, \frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}.$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4]
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9]

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 \(\left(2, \frac{\pi}{3}\right)\).(5)

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 the diagram above, is bounded by the initial line, the line \(\theta = \frac{\pi}{2}\), the arc \(AP\) of \(C\) and the line \(PN\).

1. Calculate the exact area of \(R\). (8)

\includegraphics{figure_8}

\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3\sqrt{\cos(2\theta)}, \quad \text{where } -\frac{\pi}{4} < \theta \leq \frac{\pi}{4}, \quad \frac{3\pi}{4} < \theta \leq \frac{5\pi}{4}$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4 marks]
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9 marks]