1.03g Parametric equations: of curves and conversion to cartesian

1. The curve \(C\) has equation

$$x = 3 \tan \left( y - \frac { \pi } { 6 } \right) \quad x \in \mathbb { R } \quad - \frac { \pi } { 3 } < y < \frac { 2 \pi } { 3 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { x ^ { 2 } + b }$$ where \(a\) and \(b\) are integers to be found. The point \(P\) with \(y\) coordinate \(\frac { \pi } { 3 }\) lies on \(C\).
   Given that the tangent to \(C\) at \(P\) crosses the \(x\)-axis at the point \(Q\).
2. find, in simplest form, the exact \(x\) coordinate of \(Q\).

1. The point \(P\) lies on the curve with equation

$$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,

1. find the exact value of \(p\) The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
2. Use calculus to find the coordinates of \(A\).

11. The curve \(C\) has parametric equations $$x = 10 \cos 2 t , \quad y = 6 \sin t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$ The point \(A\) with coordinates \(( 5,3 )\) lies on \(C\).

1. Find the value of \(t\) at the point \(A\).
2. Show that an equation of the normal to \(C\) at \(A\) is $$3 y = 10 x - 41$$ The normal to \(C\) at \(A\) cuts \(C\) again at the point \(B\).
3. Find the exact coordinates of \(B\).

9. \begin{figure}[h] \end{figure} The curve \(C\) has parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 4 } { t ^ { 2 } } \quad t > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = \ln 3\) and \(x = \ln 5\)

1. Show that the area of \(R\) is given by the integral $$\int _ { 1 } ^ { 3 } \frac { 4 } { t ^ { 2 } ( t + 2 ) } \mathrm { d } t$$
2. Hence find an exact value for the area of \(R\). Write your answer in the form ( \(a + \ln b\) ), where \(a\) and \(b\) are rational numbers.
3. Find a cartesian equation of the curve \(C\) in the form \(y = \mathrm { f } ( x )\).

13. A curve \(C\) has parametric equations $$x = 6 \cos 2 t , \quad y = 2 \sin t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 }$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t\), giving the exact value of the constant \(\lambda\).
2. Find an equation of the normal 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 simplified surds. The cartesian equation for the curve \(C\) can be written in the form $$x = f ( y ) , \quad - k < y < k$$ where \(\mathrm { f } ( y )\) is a polynomial in \(y\) and \(k\) is a constant.
3. Find \(\mathrm { f } ( y )\).
4. State the value of \(k\).

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.

8. A curve has parametric equations $$x = t ^ { 2 } - t , \quad y = \frac { 4 t } { 1 - t } \quad t \neq 1$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer as a simplified fraction.
2. Find an equation for the tangent to the curve at the point \(P\) where \(t = - 1\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The tangent to the curve at \(P\) cuts the curve at the point \(Q\).
3. Use algebra to find the coordinates of \(Q\).

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan t , \quad y = 2 \sin ^ { 2 } t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The finite region \(S\), shown shaded 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.

1. Show that the volume of the solid of revolution formed is given by $$4 \pi \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \tan ^ { 2 } t - \sin ^ { 2 } t \right) \mathrm { d } t$$
2. Hence use integration to find the exact value for this volume.

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = t ^ { 2 } + 2 t , \quad y = t ^ { 3 } - 9 t , \quad t \in \mathbb { R }$$ The curve cuts the \(x\)-axis at the origin and at the points \(A\) and \(B\) as shown in Figure 3 .

1. Find the coordinates of point \(A\) and show that point \(B\) has coordinates ( 15,0 ).
2. Show that the equation of the tangent to the curve at \(B\) is \(9 x - 4 y - 135 = 0\) The tangent to the curve at \(B\) cuts the curve again at the point \(X\).
3. Find the coordinates of \(X\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

14. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \cos ^ { 3 } \theta , \quad y = 6 \sin ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Given that the point \(P\) lies on \(C\) and has parameter \(\theta = \frac { \pi } { 3 }\)

1. find the coordinates of \(P\). The line \(l\) is the normal to \(C\) at \(P\).
2. Show that an equation of \(l\) is \(y = x + 3.5\) The finite region \(S\), shown shaded in Figure 6, is bounded by the curve \(C\), the line \(l\), the \(y\)-axis and the \(x\)-axis.
3. Show that the area of \(S\) is given by $$4 + 144 \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \sin \theta \cos ^ { 2 } \theta - \sin \theta \cos ^ { 4 } \theta \right) d \theta$$
4. Hence, by integration, find the exact area of \(S\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

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2. A curve \(C\) has parametric equations $$x = \frac { 3 } { 2 } t - 5 , \quad y = 4 - \frac { 6 } { t } \quad t \neq 0$$

1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(t = 3\), giving your answer as a fraction in its simplest form.
2. Show that a cartesian equation of \(C\) can be expressed in the form $$y = \frac { a x + b } { x + 5 } \quad x \neq k$$ where \(a , b\) and \(k\) are integers to be found.

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 \(\theta = \frac { \pi } { 6 }\)

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \frac { 20 t } { 2 t + 1 } \quad y = t ( t - 4 ) , \quad t > 0$$ The curve cuts the \(x\)-axis at the point \(P\).

1. Find the \(x\) coordinate of \(P\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - A ) ( 2 t + 1 ) ^ { 2 } } { B }\) where \(A\) and \(B\) are constants to be found.
3. 1. Make \(t\) the subject of the formula $$x = \frac { 20 t } { 2 t + 1 }$$
   2. Hence find a cartesian equation of the curve \(C\). Write your answer in the form $$y = \mathrm { f } ( x ) , \quad 0 < x < k$$ where \(\mathrm { f } ( x )\) is a single fraction and \(k\) is a constant to be found.

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 7 t ^ { 2 } - 5 , \quad y = t \left( 9 - t ^ { 2 } \right) , \quad t \in \mathbb { R }$$

1. Find an equation of the tangent to \(C\) at the point where \(t = 1\) Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\), as shown in Figure 3
2. 1. Find the \(x\) coordinate of the point \(A\).
   2. Find the \(x\) coordinate of the point \(B\). The region \(R\), shown shaded in Figure 3, is enclosed by the loop of the curve \(C\).
3. Use integration to find the area of \(R\).

10. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ 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. \includegraphics[max width=\textwidth, alt={}, center]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-39_61_29_2608_1886}

3. The curve \(C\) has equation $$x = 2 \sin y .$$

1. Show that the point \(P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)\) lies on \(C\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }\) at \(P\).
3. Find an equation of the normal to \(C\) at \(P\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are exact constants.

4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.

5. \begin{figure}[h] \end{figure} The curve shown in Figure 1 has equation $$x = 3 \sin y + 3 \cos y , \quad - \frac { \pi } { 4 } < y < \frac { \pi } { 4 }$$

1. Express the equation of the curve in the form \(x = R \sin ( y + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
2. Find the coordinates of the point on the curve where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(\frac { 1 } { 2 }\). Give your answers to 3 decimal places.

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(x = \left( 9 + 16 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).
The curve crosses the \(x\)-axis at the point \(A\).

1. State the coordinates of \(A\).
2. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\), in terms of \(y\).
3. Find an equation of the tangent to the curve at \(A\).

4. The curve \(C\) is defined by the parametric equations $$x = \frac { 1 } { t } + 2 \quad y = \frac { 1 - 2 t } { 3 + t } \quad t > 0$$

1. Show that the equation of \(C\) can be written in the form \(y = \mathrm { g } ( x )\) where g is the function $$\mathrm { g } ( x ) = \frac { a x + b } { c x + d } \quad x > k$$ where \(a , b , c , d\) and \(k\) are integers to be found.
2. Hence, or otherwise, state the range of g .
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9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = \tan \theta \quad y = 2 \sin 2 \theta \quad \theta \geqslant 0$$ The finite region, shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \sqrt { 3 }\) The region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the exact volume of this solid of revolution is given by $$\int _ { 0 } ^ { k } p ( 1 - \cos 2 \theta ) d \theta$$ where \(p\) and \(k\) are constants to be found.
2. Hence find, by algebraic integration, the exact volume of this solid of revolution.

2. The curve \(C\) has parametric equations $$x = \frac { t ^ { 4 } } { 2 t + 1 } \quad y = \frac { t ^ { 3 } } { 2 t + 1 } \quad t > 0$$

1. Write down \(\frac { x } { y }\) in terms of \(t\), giving your answer in simplest form.
2. Hence show that all points on \(C\) satisfy the equation $$x ^ { 3 } - 2 x y ^ { 3 } - y ^ { 4 } = 0$$

3. The curve \(C\) has parametric equations $$x = 3 + 2 \sin t \quad y = \frac { 6 } { 7 + \cos 2 t } \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Show that \(C\) has Cartesian equation $$y = \frac { 12 } { ( 7 - x ) ( 1 + x ) } \quad p \leqslant x \leqslant q$$ where \(p\) and \(q\) are constants to be found.
2. Hence, find a Cartesian equation for \(C\) in the form $$y = \frac { a } { x + b } + \frac { c } { x + d } \quad p \leqslant x \leqslant q$$ where \(a , b , c\) and \(d\) are constants.

1. A set of points \(P ( x , y )\) is defined by the parametric equations

$$x = \frac { t - 1 } { 2 t + 1 } \quad y = \frac { 6 } { 2 t + 1 } \quad t \neq - \frac { 1 } { 2 }$$

1. Show that all points \(P ( x , y )\) lie on a straight line.
2. Hence or otherwise, find the \(x\) coordinate of the point of intersection of this line and the line with equation \(y = x + 12\)