1.03g Parametric equations: of curves and conversion to cartesian

16. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations \(x = - 3 + 6 \sin \theta , \quad y = 9 \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 4 }\) where \(\theta\) is a parameter.
a. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\) The line \(l\) is normal to \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 6 }\) b. Show that an equation for \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 9 } { 2 }$$ c. The cartesian equation for the curve \(C\) can be written in the form $$y = a - \frac { 1 } { 2 } ( x + b ) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found. The straight line with equation $$y = \frac { 1 } { 3 } x + k$$ where \(k\) is a constant intersects \(C\) at two distinct points.
d. Find the range of possible values for \(k\).

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

$$x = 3 + 2 \sin t , \quad y = 4 + 2 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$

1. Show that all points on \(C\) satisfy \(y = 6 - ( x - 3 ) ^ { 2 }\)
2. 1. Sketch the curve \(C\).
   2. Explain briefly why \(C\) does not include all points of \(y = 6 - ( x - 3 ) ^ { 2 } , \quad x \in \mathbb { R }\) The line with equation \(x + y = k\), where \(k\) is a constant, intersects \(C\) at two distinct points.
3. State the range of values of \(k\), writing your answer in set notation.

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \sin ^ { 2 } t \quad y = 2 \sin 2 t + 3 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 6, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 4\)

1. Show that the area of \(R\) is given by $$\int _ { 0 } ^ { a } \left( 8 - 8 \cos 4 t + 48 \sin ^ { 2 } t \cos t \right) \mathrm { d } t$$ where \(a\) is a constant to be found.
2. Hence, using algebraic integration, find the exact area of \(R\).

12. \begin{figure}[h] \end{figure} Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground. The vertical height, \(H\) metres, of the ball above the ground has been plotted against the horizontal distance travelled, \(x\) metres, measured from where the ball was hit. The ball is modelled as a particle travelling in a vertical plane above horizontal ground.
Given that the ball

• is hit from a point on the top of a platform of vertical height 3 m above the ground
• reaches its maximum vertical height after travelling a horizontal distance of 90 m
• is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that \(H\) is modelled as a quadratic function in \(x\)

1. find \(H\) in terms of \(x\)
2. Hence find, according to the model,
   1. the maximum vertical height of the ball above the ground,
   2. the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.
3. The possible effects of wind or air resistance are two limitations of the model. Give one other limitation of this model.

4. Given that \(a\) is a positive constant and $$\int _ { a } ^ { 2 a } \frac { t + 1 } { t } \mathrm {~d} t = \ln 7$$ show that \(a = \ln k\), where \(k\) is a constant to be found.

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

$$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$

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 the 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.

14. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) , y = 2 \sin t , \quad 0 < t \leqslant 2 \pi$$ Show that a Cartesian equation of \(C\) can be written in the form $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be found.

4. \begin{figure}[h] \end{figure} The curve \(C _ { 1 }\) with parametric equations $$x = 10 \cos t , \quad y = 4 \sqrt { 2 } \sin t , \quad 0 \leqslant t < 2 \pi$$ meets the circle \(C _ { 2 }\) with equation $$x ^ { 2 } + y ^ { 2 } = 66$$ at four distinct points as shown in Figure 2.
Given that one of these points, \(S\), lies in the 4th quadrant, find the Cartesian coordinates of \(S\).

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 \(k\).

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

$$x = t ^ { 2 } + 6 t - 16 \quad y = 6 \ln ( t + 3 ) \quad t > - 3$$

1. Show that a Cartesian equation for \(C\) is $$y = A \ln ( x + B ) \quad x > - B$$ where \(A\) and \(B\) are integers to be found. The curve \(C\) cuts the \(y\)-axis at the point \(P\)
2. Show that the equation of the tangent to \(C\) at \(P\) can be written in the form $$a x + b y = c \ln 5$$ where \(a\), \(b\) and \(c\) are integers to be found.

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = ( t + 3 ) ^ { 2 } \quad y = 1 - t ^ { 3 } \quad - 2 \leqslant t \leqslant 1$$ The point \(P\) with coordinates \(( 4,2 )\) lies on \(C\).

1. Using parametric differentiation, show that the tangent to \(C\) at \(P\) has equation $$3 x + 4 y = 20$$ The curve \(C\) is used to model the profile of a slide at a water park.
   Units are in metres, with \(y\) being the height of the slide above water level.
2. Find, according to the model, the greatest height of the slide above water level.

12. \begin{figure}[h] \end{figure} The curve shown in Figure 3 has parametric equations $$x = 6 \sin t \quad y = 5 \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis. \begin{enumerate}[label=(\alph*)] \item

1. Show that the area of \(R\) is given by \(\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \sin t \cos ^ { 2 } t \mathrm {~d} t\)
2. Hence show, by algebraic integration, that the area of \(R\) is exactly 20 \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_451_570_1416_742} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { t + 1 } , \quad t > - \frac { 2 } { 3 }$$

1. State the domain of values of \(x\) for the curve \(C\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = \ln 2\), the \(x\)-axis and the line with equation \(x = \ln 4\)
2. Use calculus to show that the area of \(R\) is \(\ln \left( \frac { 3 } { 2 } \right)\).

12 Fig. 12 shows a curve C with parametric equations \(x = 4 t ^ { 2 } , y = 4 t\). The point P , with parameter \(t\), is a general point on the curve. Q is the point on the line \(x + 4 = 0\) such that PQ is parallel to the \(x\)-axis. R is the point \(( 4,0 )\). \begin{figure}[h] \end{figure}

1. Show algebraically that P is equidistant from Q and R .
2. Find a cartesian equation of C .

8 A particle moves in the \(x - y\) plane so that its position at time \(t\) s is given by \(x = t ^ { 3 } - 8 t , y = t ^ { 2 }\) for \(- 3.5 < t < 3.5\). The units of distance are metres. The graph shows the path of the particle and the direction of travel at the point \(\mathrm { P } ( 8,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-07_492_924_415_242}

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) in terms of \(t\).
2. Hence show that the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at P is - 1 .
3. Find the time at which the particle is travelling in the direction opposite to that at P .
4. Find the cartesian equation of the path, giving \(x ^ { 2 }\) as a function of \(y\).

11 Fig. 11 shows the curve with parametric equations $$x = 2 \cos \theta , y = \sin \theta , 0 \leq \theta \leq 2 \pi .$$ The point P has parameter \(\frac { 1 } { 4 } \pi\). The tangent at P to the curve meets the axes at A and B . \begin{figure}[h] \end{figure}

1. Show that the equation of the line AB is \(x + 2 y = 2 \sqrt { 2 }\).
2. Determine the area of the triangle AOB .

6 The parametric equations of a circle are \(x = 2 \cos \theta - 3\) and \(y = 2 \sin \theta + 1\).
Determine the cartesian equation of the circle in the form \(( \mathrm { x } - \mathrm { a } ) ^ { 2 } + ( \mathrm { y } - \mathrm { b } ) ^ { 2 } = \mathrm { k }\), where \(a , b\) and \(k\) are integers.

7 The parametric equations of a circle are \(x = 7 + 5 \cos \theta , \quad y = 5 \sin \theta - 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).

1. Find a cartesian equation of the circle.
2. State the coordinates of the centre of the circle. Answer all the questions.
   Section B (77 marks)

8 A curve has parametric equations \(x = \frac { t } { 1 + t ^ { 3 } } , y = \frac { t ^ { 2 } } { 1 + t ^ { 3 } }\), where \(t \neq - 1\).

1. In this question you must show detailed reasoning. Determine the gradient of the curve at the point where \(t = 1\).
2. Verify that the cartesian equation of the curve is \(x ^ { 3 } + y ^ { 3 } = x y\).

12 The diagram shows the curve with parametric equations \(x = \sin 2 \theta + 2 , y = 2 \cos \theta + \cos 2 \theta\), for \(0 \leqslant \theta < 2 \pi\). \includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-10_771_673_397_239}

1. In this question you must show detailed reasoning. Determine the exact coordinates of all the stationary points on the curve.
2. Write down the equation of the line of symmetry of the curve.

1 A curve is given by \(x = t ^ { 2 } - 2 \ln t , y = 4 t\) for \(t > 0\). When the arc of the curve between the points where \(t = 1\) and \(t = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
Given that \(A = k \pi\), where \(k\) is an integer,

• write down an integral which gives \(A\) and
• find the value of \(k\).

4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$

1. 1. Find the gradient of the curve at the point where \(t = 0\).
   2. Find an equation of the tangent to the curve at the point where \(t = 0\).
2. Show that the cartesian equation of the curve can be written in the form $$y = \frac { x ^ { 2 } } { k } - \frac { k } { x ^ { 2 } }$$ where \(k\) is an integer.

5 A curve is defined by the parametric equations $$x = 8 t ^ { 2 } - t , \quad y = \frac { 3 } { t }$$

1. Show that the cartesian equation of the curve can be written as \(x y ^ { 2 } + 3 y = k\), stating the value of the integer \(k\).
   (2 marks)
2. 1. Find an equation of the tangent to the curve at the point \(P\), where \(t = \frac { 1 } { 4 }\).
   2. Verify that the tangent at \(P\) intersects the curve when \(x = \frac { 3 } { 2 }\).

4

1. A curve is defined by the equation \(x ^ { 2 } - y ^ { 2 } = 8\).
   1. Show that at any point \(( p , q )\) on the curve, where \(q \neq 0\), the gradient of the curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { q }\).
      (2 marks)
   2. Show that the tangents at the points \(( p , q )\) and \(( p , - q )\) intersect on the \(x\)-axis.
      (4 marks)
2. Show that \(x = t + \frac { 2 } { t } , y = t - \frac { 2 } { t }\) are parametric equations of the curve \(x ^ { 2 } - y ^ { 2 } = 8\).
   (2 marks)

4

1. A curve is defined by the parametric equations \(x = 3 \cos 2 \theta , y = 2 \cos \theta\).
   1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }\), where \(k\) is an integer.
   2. Find an equation of the normal to the curve at the point where \(\theta = \frac { \pi } { 3 }\).
2. Find the exact value of \(\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x\).