Convert to Cartesian (tan/sec/cot/cosec identities)

3. A curve \(C\) has parametric equations $$x = \sqrt { 3 } \tan \theta , \quad y = \sec ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ The cartesian equation of \(C\) is $$y = \mathrm { f } ( x ) , \quad 0 \leqslant x \leqslant k , \quad \text { where } k \text { is a constant }$$

1. State the value of \(k\).
2. Find \(\mathrm { f } ( x )\) in its simplest form.
3. Hence, or otherwise, find the gradient of the curve at the point where \(\theta = \frac { \pi } { 6 }\)

1. A curve has parametric equations

$$x = 2 \cot t , \quad y = 2 \sin ^ { 2 } t , \quad 0 < t \leqslant \frac { \pi } { 2 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of the parameter \(t\).
2. Find an equation of the tangent to the curve at the point where \(t = \frac { \pi } { 4 }\).
3. Find a cartesian equation of the curve in the form \(y = \mathrm { f } ( x )\). State the domain on which the curve is defined.

6. A curve has parametric equations $$x = \tan ^ { 2 } t , \quad y = \sin t , \quad 0 < t < \frac { \pi } { 2 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). You need not simplify your answer.
2. Find an equation of the tangent to the curve at the point where \(t = \frac { \pi } { 4 }\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be determined.
3. Find a cartesian equation of the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
   \section*{LU}

4. \includegraphics[max width=\textwidth, alt={}, center]{c7b867af-0730-459e-9c76-15eb07b9e476-1_465_976_1539_388} The diagram shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$

1. Find a cartesian equation for the curve. The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\).
2. Using integration, with the substitution \(x = \tan u\), find the area of the shaded region.

9. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$

1. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
2. find a cartesian equation for the curve.
3. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).

1 A curve has parametric equations \(x = \sec \theta , y = 2 \tan \theta\).

1. Given that the derivative of \(\sec \theta\) is \(\sec \theta \tan \theta\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta\).
2. Verify that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4\). Fig. 5 shows the region enclosed by the curve and the line \(x = 2\). This region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1e788cb0-36b0-42a9-9e0c-077022d410ae-1_556_867_580_588} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure}
3. Find the volume of revolution produced, giving your answer in exact form.

5 A curve has parametric equations \(x = \sec \theta , y = 2 \tan \theta\).

1. Given that the derivative of \(\sec \theta\) is \(\sec \theta \tan \theta\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta\).
2. Verify that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4\). Fig. 5 shows the region enclosed by the curve and the line \(x = 2\). This region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-02_545_853_1738_607} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure}
3. Find the volume of revolution produced, giving your answer in exact form.

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \tan t + 1 \quad y = 2 \sec ^ { 2 } t + 3 \quad - \frac { \pi } { 4 } \leqslant t \leqslant \frac { \pi } { 3 }$$ The line \(l\) is the normal to \(C\) at the point \(P\) where \(t = \frac { \pi } { 4 }\)

1. Using parametric differentiation, show that an equation for \(l\) is $$y = - \frac { 1 } { 2 } x + \frac { 17 } { 2 }$$
2. Show that all points on \(C\) satisfy the equation $$y = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 } + 5$$ The straight line with equation $$y = - \frac { 1 } { 2 } x + k \quad \text { where } k \text { is a constant }$$ intersects \(C\) at two distinct points.
3. Find the range of possible values for \(k\).

12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).

1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).