Convert to Cartesian (polynomial/rational)

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.

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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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 $$x ^ { 3 } - 2 x y ^ { 3 } - y ^ { 4 } = 0$$

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 \(y = x + 12\)

4 A curve is defined by parametric equations $$x = \frac { 1 } { t } - 1 , y = \frac { 2 + t } { 1 + t }$$ Show that the cartesian equation of the curve is \(y = \frac { 3 + 2 x } { 2 + x }\).

5 A ball is thrown towards a hedge. Its position relative to the point from which it was thrown is given by the parametric equations $$x = 8 t , y = 10 t - 5 t ^ { 2 }$$

1. Find the cartesian equation of the trajectory of the ball.
2. The ball just clears the hedge. What can you say about the height of the hedge?

6. A curve has parametric equations $$x = \frac { t } { 2 - t } , \quad y = \frac { 1 } { 1 + t } , \quad - 1 < t < 2$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \left( \frac { 2 - t } { 1 + t } \right) ^ { 2 }\).
2. Find an equation for the normal to the curve at the point where \(t = 1\).
3. Show that the cartesian equation of the curve can be written in the form $$y = \frac { 1 + x } { 1 + 3 x }$$

8 A curve is defined by parametric equations $$x = \frac { 1 } { t } - 1 , y = \frac { 2 + t } { 1 + t }$$ Show that the cartesian equation of the curve is \(y = \frac { 3 + 2 x } { 2 + x }\).

5 A curve is defined parametrically by the equations $$x = \frac { 1 } { 1 + t } , \quad y = \frac { 1 - t } { 1 + 2 t }$$ Find \(t\) in terms of \(x\). Hence find the cartesian equation of the curve, giving your answer as simply as possible.

5 A curve has parametric equations $$x = 2 t + t ^ { 2 } , \quad y = 2 t ^ { 2 } + t ^ { 3 }$$

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and find the gradient of the curve at the point \(( 3 , - 9 )\).
2. By considering \(\frac { y } { x }\), find a cartesian equation of the curve, giving your answer in a form not involving fractions.

7 The parametric equations of a curve are \(x = \frac { t + 2 } { t + 1 } , y = \frac { 2 } { t + 3 }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\).
2. Find the cartesian equation of the curve, giving your answer in a form not involving fractions.

2 A curve is defined parametrically by the equations $$x = \frac { 1 } { 1 + t } , \quad y = \frac { 1 - t } { 1 + 2 t }$$ Find \(t\) in terms of \(x\). Hence find the cartesian equation of the curve, giving your answer as simply as possible.

10 A curve has parametric equations \(x = t + \frac { 2 } { t }\) and \(y = t - \frac { 2 } { t }\), for \(t \neq 0\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
2. Explain why the curve has no stationary points.
3. By considering \(x + y\), or otherwise, find a cartesian equation of the curve, giving your answer in a form not involving fractions or brackets.

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

$$x = \frac { t ^ { 2 } + 5 } { t ^ { 2 } + 1 } \quad y = \frac { 4 t } { t ^ { 2 } + 1 } \quad t \in \mathbb { R }$$ Show that all points on \(C\) satisfy $$( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$$

5. A curve \(C\) has parametric equations $$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$ Show that the Cartesian equation of the curve \(C\) can be written in the form $$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$ where \(a\) and \(b\) are integers to be found.

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 \(y\).

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 \(x ^ { 3 } + y ^ { 3 } = x y\).

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 \(x = \frac { 3 } { 2 }\).

5 A curve is defined by the parametric equations \(x = 2 t + \frac { 1 } { t ^ { 2 } } , \quad y = 2 t - \frac { 1 } { t ^ { 2 } }\).

1. At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).
   1. Find the coordinates of \(P\).
   2. Find an equation of the tangent to the curve at \(P\).
2. Show that the cartesian equation of the curve can be written as $$( x - y ) ( x + y ) ^ { 2 } = k$$ where \(k\) is an integer.

5 A curve is defined by the parametric equations $$x = 2 t + \frac { 1 } { t } , \quad y = \frac { 1 } { t } , \quad t \neq 0$$

1. Find the coordinates of the point on the curve where \(t = \frac { 1 } { 2 }\).
2. Show that the cartesian equation of the curve can be written as $$x y - y ^ { 2 } = 2$$
3. Show that the gradient of the curve at the point \(( 3,2 )\) is 2 .

7 A curve is given parametrically by \(x = 3 t , y = 1 + t ^ { 3 }\) where \(t\) is any real number.

1. Show that a cartesian equation for this curve is given by \(y = 1 + \frac { 1 } { 27 } x ^ { 3 }\). A second curve is given by \(y = x ^ { 2 } + 4 x - 19\).
2. Given that the curves intersect at the point \(( 3,2 )\), find the coordinates of all the other points of intersection between the two curves.

A curve \(C\) has parametric equations $$x = \frac{t}{t-3}, \quad y = \frac{1}{t} + 2, \quad t \in \mathbb{R}, \quad t > 3$$ Show that all points on \(C\) lie on the curve with Cartesian equation $$y = \frac{ax - 1}{bx}$$ where \(a\) and \(b\) are constants to be found. [3]