1.03h Parametric equations: in modelling contexts

5 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-2_512_903_1302_621} The diagram represents a metal plate \(O A B C\), consisting of a sector \(O A B\) of a circle with centre \(O\) and radius \(r\), together with a triangle \(O C B\) which is right-angled at \(C\). Angle \(A O B = \theta\) radians and \(O C\) is perpendicular to \(O A\).

1. Find an expression in terms of \(r\) and \(\theta\) for the perimeter of the plate.
2. For the case where \(r = 10\) and \(\theta = \frac { 1 } { 5 } \pi\), find the area of the plate.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a design for a scraper blade. The blade \(A O B C D A\) consists of an isosceles triangle \(C O D\) joined along its equal sides to sectors \(O B C\) and \(O D A\) of a circle with centre \(O\) and radius 8 cm . Angles \(A O D\) and \(B O C\) are equal. \(A O B\) is a straight line and is parallel to the line \(D C . D C\) has length 7 cm .

1. Show that the angle \(C O D\) is 0.906 radians, correct to 3 significant figures.
2. Find the perimeter of \(A O B C D A\), giving your answer to 3 significant figures.
3. Find the area of \(A O B C D A\), giving your answer to 3 significant figures.

13. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 4 has parametric equations $$x = 1 + \sqrt { 3 } \tan \theta , \quad y = 5 \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The curve \(C\) crosses the \(y\)-axis at \(A\) and has a minimum turning point at \(B\), as shown in Figure 4.

1. Find the exact coordinates of \(A\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \sin \theta\), giving the exact value of the constant \(\lambda\).
3. Find the coordinates of \(B\).
4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y = k \sqrt { \left( x ^ { 2 } - 2 x + 4 \right) }$$ where \(k\) is a simplified surd to be found.

11. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 3 has parametric equations $$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leqslant t \leqslant 2 \pi$$ The curve \(C\) meets the \(x\)-axis at the origin and at the points \(A\) and \(B\), as shown in Figure 3 .

1. Write down the coordinates of \(A\) and \(B\).
2. Find the values of \(t\) at which the curve passes through the origin.
3. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), and hence find the gradient of the curve when \(t = \frac { \pi } { 6 }\)
4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y ^ { 2 } = a x ^ { 2 } \left( b - x ^ { 2 } \right)$$ where \(a\) and \(b\) are integers to be determined.

1. The hyperbola \(H\) has Cartesian equation \(x y = 25\)

The parabola \(P\) has parametric equations \(x = 10 t ^ { 2 } , y = 20 t\) The hyperbola \(H\) intersects the parabola \(P\) at the point \(A\)

1. Use algebra to determine the coordinates of \(A\) The point \(B\) with coordinates \(( 10,20 )\) lies on \(P\)
2. Find an equation for the normal to \(P\) at \(B\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined.
3. Use algebra to determine, in simplest form, the exact coordinates of the points where this normal intersects the hyperbola \(H\) (6)

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 8 x\). The point \(P\) lies on \(C\), where \(y > 0\), and the point \(Q\) lies on \(C\), where \(y < 0\) The line segment \(P Q\) is parallel to the \(y\)-axis. Given that the distance \(P Q\) is 12 ,

1. write down the \(y\)-coordinate of \(P\),
2. find the \(x\)-coordinate of \(P\). Figure 1 shows the point \(S\) which is the focus of \(C\).
   The line \(l\) passes through the point \(P\) and the point \(S\).
3. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

9. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 4 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l\) is the normal to the ellipse at the point \(P\).

1. Show that an equation for \(l\) is $$5 x \sin \theta - 4 y \cos \theta = 9 \sin \theta \cos \theta$$ The point \(F\) is the focus of \(E\) that lies on the positive \(x\)-axis.
2. Determine the coordinates of \(F\). The line \(l\) crosses the \(x\)-axis at the point \(Q\).
3. Show that $$\frac { | Q F | } { | P F | } = e$$ where \(e\) is the eccentricity of \(E\).
   

   END

3

1. 1. Given that \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }\). The arc of the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln a\) (where \(a > 1\) ) is denoted by \(C\).
   2. Show that the length of \(C\) is \(\frac { a - 1 } { \sqrt { a } }\).
   3. Find the area of the surface formed when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. An ellipse has parametric equations \(x = 2 \cos \theta , y = \sin \theta\) for \(0 \leqslant \theta < 2 \pi\).
   1. Show that the normal to the ellipse at the point with parameter \(\theta\) has equation $$y = 2 x \tan \theta - 3 \sin \theta$$
   2. Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation $$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
   3. Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
      (A) at the point \(( 2,0 )\),
      (B) at the point \(( 0,1 )\).

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 \(k\).