1.03g Parametric equations: of curves and conversion to cartesian

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \sec t \quad y = \sqrt { 3 } \tan \left( t + \frac { \pi } { 3 } \right) \quad \frac { \pi } { 6 } < t < \frac { \pi } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\)
2. Find an equation for the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
3. Show that all points on \(C\) satisfy the equation $$y = \frac { A x ^ { 2 } + B \sqrt { 3 x ^ { 2 } - 3 } } { 4 - 3 x ^ { 2 } }$$ where \(A\) and \(B\) are constants to be found.

8. \begin{figure}[h] \end{figure} The curve shown in Figure 2 has parametric equations $$x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leqslant t \leqslant 2 \pi$$

1. Show that the curve crosses the \(x\)-axis where \(t = \frac { \pi } { 3 }\) and \(t = \frac { 5 \pi } { 3 }\). The finite region \(R\) is enclosed by the curve and the \(x\)-axis, as shown shaded in Figure 2.
2. Show that the area of \(R\) is given by the integral $$\int _ { \frac { \pi } { 3 } } ^ { \frac { 5 \pi } { 3 } } ( 1 - 2 \cos t ) ^ { 2 } \mathrm {~d} t$$
3. Use this integral to find the exact value of the shaded area.

1. A curve has parametric equations

$$x = 7 \cos t - \cos 7 t , y = 7 \sin t - \sin 7 t , \quad \frac { \pi } { 8 } < t < \frac { \pi } { 3 }$$

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 normal to the curve at the point where \(t = \frac { \pi } { 6 }\). Give your answer in its simplest exact form.

7. \begin{figure}[h] \end{figure} The curve \(C\) has parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { ( t + 1 ) } , \quad t > - 1$$ The finite region \(R\) between the curve \(C\) and the \(x\)-axis, bounded by the lines with equations \(x = \ln 2\) and \(x = \ln 4\), is shown shaded in Figure 3.

1. Show that the area of \(R\) is given by the integral $$\int _ { 0 } ^ { 2 } \frac { 1 } { ( t + 1 ) ( t + 2 ) } \mathrm { d } t$$
2. Hence find an exact value for this area.
3. Find a cartesian equation of the curve \(C\), in the form \(y = \mathrm { f } ( x )\).
4. State the domain of values for \(x\) for this curve. \(\_\_\_\_\)}

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 5 t ^ { 2 } - 4 , \quad y = t \left( 9 - t ^ { 2 } \right)$$ The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\).

1. Find the \(x\)-coordinate at the point \(A\) and the \(x\)-coordinate at the point \(B\). The region \(R\), as shown shaded in Figure 2, is enclosed by the loop of the curve.
2. Use integration to find the area of \(R\).
   \section*{LU}

1. The curve \(C\) has parametric equations

$$x = \ln t , \quad y = t ^ { 2 } - 2 , \quad t > 0$$ Find

1. an equation of the normal to \(C\) at the point where \(t = 3\),
2. a cartesian equation of \(C\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a3ece8a8-8107-4c3a-a6a9-c19b5e35ec5a-10_579_759_740_571} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The finite area \(R\), shown in Figure 1, is bounded by \(C\), the \(x\)-axis, the line \(x = \ln 2\) and the line \(x = \ln 4\). The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Use calculus to find the exact volume of the solid generated.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \sin \left( t + \frac { \pi } { 6 } \right) , \quad y = 3 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the coordinates of all the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

7. The curve \(C\) has parametric equations $$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$ where \(t\) is a parameter.

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\) The line \(l\) is a normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line \(l\) intersects the curve \(C\) again at the point \(Q\).
3. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-23_106_63_2595_1882}

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.

4. \begin{figure}[h] \end{figure} The curve shown in Figure 2 has parametric equations $$x = \sin t , y = \sin \left( t + \frac { \pi } { 6 } \right) , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 }$$

1. Find an equation of the tangent to the curve at the point where \(t = \frac { \pi } { 6 }\).
2. Show that a cartesian equation of the curve is $$y = \frac { \sqrt { } 3 } { 2 } x + \frac { 1 } { 2 } \sqrt { } \left( 1 - x ^ { 2 } \right) , \quad - 1 < x < 1$$

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}

8. \begin{figure}[h] \end{figure} Figure 3 shows the curve \(C\) with parametric equations $$x = 8 \cos t , \quad y = 4 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 } .$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,2 \sqrt { } 3 )\).

1. Find the value of \(t\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is \(y = - x \sqrt { 3 } + 6 \sqrt { 3 }\). The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 4\), as shown shaded in Figure 3.
3. Show that the area of \(R\) is given by the integral \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 2 } t \cos t \mathrm {~d} t\).
4. Use this integral to find the area of \(R\), giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are constants to be determined.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with parametric equations $$x = 2 \cos 2 t , \quad y = 6 \sin t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Find the gradient of the curve at the point where \(t = \frac { \pi } { 3 }\).
2. Find a cartesian equation of the curve in the form $$y = \mathrm { f } ( x ) , \quad - k \leqslant x \leqslant k$$ stating the value of the constant \(k\).
3. Write down the range of \(\mathrm { f } ( x )\).

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { } 3 , \frac { 1 } { 2 } \sqrt { } 3 \right)\).

1. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
2. Show that \(Q\) has coordinates \(( k \sqrt { } 3,0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { } 3\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { } 3 + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants. Question 7 continued
   8. (a) Find \(\int ( 4 y + 3 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} y\) (b) Given that \(y = 1.5\) at \(x = - 2\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { } ( 4 y + 3 ) } { x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = ( \sqrt { } 3 ) \sin 2 t , \quad y = 4 \cos ^ { 2 } t , \quad 0 \leqslant t \leqslant \pi$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k ( \sqrt { } 3 ) \tan 2 t\), where \(k\) is a constant to be determined.
2. Find an equation of the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
3. Find a cartesian equation of \(C\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$

1. Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
2. Show that the cartesian equation of \(C\) may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of \(a\) and \(b\).
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Use calculus to find the exact value of the volume of the solid of revolution. \section*{Question 7 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Let \(x\) be the mass of waste products, in kg , at time \(t\) minutes after the start of the experiment. It is known that at time \(t\) minutes, the rate of increase of the mass of waste products, in kg per minute, is \(k\) times the mass of unburned fuel remaining, where \(k\) is a positive constant. The differential equation connecting \(x\) and \(t\) may be written in the form $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( M - x ) , \text { where } M \text { is a constant. }$$
   1. Explain, in the context of the problem, what \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(M\) represent. Given that initially the mass of waste products is zero,
   2. solve the differential equation, expressing \(x\) in terms of \(k , M\) and \(t\). Given also that \(x = \frac { 1 } { 2 } M\) when \(t = \ln 4\),
   3. find the value of \(x\) when \(t = \ln 9\), expressing \(x\) in terms of \(M\), in its simplest form. \section*{Question 8 continued}

1. A curve \(C\) has parametric equations

$$x = 2 \sin t , \quad y = 1 - \cos 2 t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(t = \frac { \pi } { 6 }\)
2. Find a cartesian equation for \(C\) in the form $$y = \mathrm { f } ( x ) , \quad - k \leqslant x \leqslant k$$ stating the value of the constant \(k\).
3. Write down the range of \(\mathrm { f } ( x )\).

8. \begin{figure}[h] \end{figure} The curve shown in Figure 3 has parametric equations $$x = t - 4 \sin t , y = 1 - 2 \cos t , \quad - \frac { 2 \pi } { 3 } \leqslant t \leqslant \frac { 2 \pi } { 3 }$$ The point \(A\), with coordinates ( \(k , 1\) ), lies on the curve. Given that \(k > 0\)

1. find the exact value of \(k\),
2. find the gradient of the curve at the point \(A\). There is one point on the curve where the gradient is equal to \(- \frac { 1 } { 2 }\)
3. Find the value of \(t\) at this point, showing each step in your working and giving your answer to 4 decimal places.
   [0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \tan t , \quad y = 5 \sqrt { 3 } \sin 2 t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( 4 \sqrt { 3 } , \frac { 15 } { 2 } \right)\).

1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer as a simplified surd. The point \(Q\) lies on the curve \(C\), where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
2. Find the exact coordinates of the point \(Q\).

1. The curve \(C\) has parametric equations

$$x = 3 t - 4 , y = 5 - \frac { 6 } { t } , \quad t > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) The point \(P\) lies on \(C\) where \(t = \frac { 1 } { 2 }\)
2. Find the equation of the tangent to \(C\) at the point \(P\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are integers to be determined.
3. Show that the cartesian equation for \(C\) can be written in the form $$y = \frac { a x + b } { x + 4 } , \quad x > - 4$$ where \(a\) and \(b\) are integers to be determined.

8. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3 \theta \sin \theta , \quad y = \sec ^ { 3 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P ( k , 8 )\) lies on \(C\), where \(k\) is a constant.

1. Find the exact value of \(k\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = k\).
2. Show that the area of \(R\) can be expressed in the form $$\lambda \int _ { \alpha } ^ { \beta } \left( \theta \sec ^ { 2 } \theta + \tan \theta \sec ^ { 2 } \theta \right) \mathrm { d } \theta$$ where \(\lambda , \alpha\) and \(\beta\) are constants to be determined.
3. Hence use integration to find the exact value of the area of \(R\).

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$ The point \(A\) lies on the curve \(C\). Given that the coordinates of \(A\) are ( \(k , 2\) ), where \(k > 0\)

1. find the exact value of \(k\), giving your answer in a fully simplified form.
2. Find the equation of the tangent to \(C\) at the point \(A\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are exact real values.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$

1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
2. Find an equation of the normal to the curve at \(P\).
3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).

4. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 12 x\) The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on \(C\), where \(p \neq 0\)

1. Show that the equation of the normal to the curve \(C\) at the point \(P\) is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve \(C\) again at the point \(Q\).
   Given that \(p = 2\) and that \(S\) is the focus of the parabola, find
2. the coordinates of the point \(Q\),
3. the area of the triangle \(P Q S\).

6. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant. The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Show that the normal to \(H\) at \(P\) has equation $$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
2. Find, in terms of \(c\) and \(p\), the coordinates of \(Q\).