1.03g Parametric equations: of curves and conversion to cartesian

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

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [4]

The point \(P\) on the curve has parameter \(t = -1\).

1. Show that the tangent to the curve at \(P\) has the equation $$x + 3y + 4 = 0.$$ [3]

The tangent to the curve at \(P\) meets the curve again at the point \(Q\).

1. Find the coordinates of \(Q\). [7]

\includegraphics{figure_1} Figure 1 shows the curve with parametric equations $$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$ where \(a\) is a positive constant.

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]

The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.

1. Show that the area of triangle \(OAB\) is \(a^2\). [6]

\includegraphics{figure_2} Figure 2 shows the curve with parametric equations $$x = t + \sin t, \quad y = \sin t, \quad 0 \leq t \leq \pi.$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis. [3]
3. Show that the region bounded by the curve and the \(x\)-axis has area 2. [6]

\includegraphics{figure_5} The diagram shows the curve with parametric equations $$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$ where \(a\) is a positive constant.

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]

The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.

1. Show that the area of triangle \(OAB\) is \(a^2\). [5]

A curve has parametric equations $$x = t^3 + 1, \quad y = \frac{2}{t}, \quad t \neq 0.$$

1. Find an equation for the normal to the curve at the point where \(t = 1\), giving your answer in the form \(y = mx + c\). [6]
2. Find a cartesian equation for the curve in the form \(y = f(x)\). [3]

A curve has parametric equations $$x = \cos 2t, \quad y = \cosec t, \quad 0 < t < \frac{\pi}{2}.$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac{1}{2}\).

1. Find the value of the parameter \(t\) at \(P\). [2]
2. Show that the tangent to the curve at \(P\) has the equation $$y = 2x + 1.$$ [5]

A curve has equation $$x^2 + 4y^2 = k^2,$$ where \(k\) is a positive constant.

1. Verify that $$x = k\cos\theta, \quad y = \frac{k}{2}\sin\theta,$$ are parametric equations for the curve. [3]
2. Hence or otherwise show that \(\frac{dy}{dx} = -\frac{x}{4y}\). [3]
3. Fig. 8 illustrates the curve for a particular value of \(k\). Write down this value of \(k\). [1]

\includegraphics{figure_8}

1. Copy Fig. 8 and on the same axes sketch the curves for \(k = 1\), \(k = 3\) and \(k = 4\). [3]

On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.

1. Explain why the path of the stream is modelled by the differential equation $$\frac{dy}{dx} = \frac{4y}{x}.$$ [2]
2. Solve this differential equation. Given that the path of the stream passes through the point (2, 1), show that its equation is \(y = \frac{x^4}{16}\). [6]

\includegraphics{figure_3} In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations $$x = 10 \cos \theta + 5 \cos 2\theta, \quad y = 10 \sin \theta + 5 \sin 2\theta, \quad (0 \leqslant \theta < 2\pi),$$ where \(x\) and \(y\) are in metres.

1. Show that \(\frac{\text{d}y}{\text{d}x} = -\frac{\cos \theta + \cos 2\theta}{\sin \theta + \sin 2\theta}\). Verify that \(\frac{\text{d}y}{\text{d}x} = 0\) when \(\theta = \frac{1}{3}\pi\). Hence find the exact coordinates of the highest point A on the path of C. [6]
2. Express \(x^2 + y^2\) in terms of \(\theta\). Hence show that $$x^2 + y^2 = 125 + 100 \cos \theta.$$ [4]
3. Using this result, or otherwise, find the greatest and least distances of C from O. [2]

You are given that, at the point B on the path vertically above O, $$2 \cos^2 \theta + 2 \cos \theta - 1 = 0.$$

1. Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. [4]

A parabola with equation \(y^2 = 4ax\), where \(a\) is a constant, is translated by the vector \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) to give the curve \(C\). The curve \(C\) passes through the point \((4, 7)\).

1. Show that \(a = 2\). [3 marks]
2. Find the values of \(k\) for which the line \(ky = x\) does not meet the curve \(C\). [6 marks]

A curve is defined parametrically by $$x = t^3 + 5, \quad y = 6t^2 - 1$$ The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).

1. Show that \(s = \int_{0}^{3} 3t\sqrt{t^2 + A} \, \text{d}t\), stating the value of the constant \(A\). [4 marks]
2. Hence show that \(s = 61\). [4 marks]

A curve has parametric equations $$x = t + a \sin t, \quad y = 1 - a \cos t,$$ where \(a\) is a positive constant.

1. Draw, on separate diagrams, sketches of the curve for \(-2\pi < t < 2\pi\) in the cases \(a = 1\), \(a = 2\) and \(a = 0.5\). By investigating other cases, state the value(s) of \(a\) for which the curve has
   1. loops,
   2. cusps. [7]
2. Suppose that the point P\((x, y)\) lies on the curve. Show that the point P\('(-x, y)\) also lies on the curve. What does this indicate about the symmetry of the curve? [3]
3. Find an expression in terms of \(a\) and \(t\) for the gradient of the curve. Hence find, in terms of \(a\), the coordinates of the turning points on the curve for \(-2\pi < t < 2\pi\) and \(a \neq 1\). [5]
4. In the case \(a = \frac{1}{2}\pi\), show that \(t = \frac{1}{3}\pi\) and \(t = \frac{5}{3}\pi\) give the same point. Find the angle at which the curve crosses itself at this point. [3]

Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}

1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] Now let the variable point P be defined by the parametric equations (*) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \to \infty\) and as \(a \to -\infty\). Show algebraically that this locus has an asymptote at \(y = -1\). On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies. [8] (The single curve made up of these two loci and including the point B is called a right strophoid.)
4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? [3]

The curve \(C\) has parametric equations $$x = 15t - t^3, \quad y = 3 - 2t^2.$$ Find the values of \(t\) at the points where the normal to \(C\) at \((14, 1)\) cuts \(C\) again. [11]

1. Given that \(y = \ln [t + \sqrt{(1 + t^2)}]\), show that \(\frac{dy}{dt} = \frac{1}{\sqrt{(1+t^2)}}\). [3]

The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{(1+t^2)}}, \quad y = \ln [t + \sqrt{(1 + t^2)}], \quad t \in \mathbb{R}.$$ A student was asked to prove that, for \(t > 0\), the gradient of the tangent to \(C\) is negative. The attempted proof was as follows: $$y = \ln \left(t + \frac{1}{x}\right)$$ $$= \ln \left(\frac{tx + 1}{x}\right)$$ $$= \ln (tx + 1) - \ln x$$ $$\therefore \frac{dy}{dx} = \frac{t}{tx + 1} - \frac{1}{x}$$ $$= \frac{\frac{t}{x}}{t + \frac{1}{x}} - \frac{1}{x}$$ $$= \frac{t\sqrt{(1+t^2)}}{t + \sqrt{(1+t^2)}} - \sqrt{(1 + t^2)}$$ $$= -\frac{(1+t^2)}{t + \sqrt{(1+t^2)}}$$ As \((1 + t^2) > 0\), and \(t + \sqrt{(1 + t^2)} > 0\) for \(t > 0\), \(\frac{dy}{dx} < 0\) for \(t > 0\).

1. 1. Identify the error in this attempt.
   2. Give a correct version of the proof. [6]
2. Prove that \(\ln [-t + \sqrt{(1 + t^2)}] = -\ln [t + \sqrt{(1 + t^2)}]\). [3]
3. Deduce that \(C\) is symmetric about the \(x\)-axis and sketch the graph of \(C\). [3]

\includegraphics{figure_5} The diagram shows the curve \(C\) with parametric equations \(x = \frac{3}{t}\), \(y = t^2 e^{-2t}\), where \(t > 0\). The maximum point on \(C\) is denoted by \(P\).

1. Determine the exact coordinates of \(P\). [4] The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
2. Show that the area of \(R\) is given by $$\int_a^b 3te^{-2t} dt,$$ where \(a\) and \(b\) are constants to be determined. [3]
3. Hence determine the exact area of \(R\). [5]

A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below. \includegraphics{figure_5} The emblem is modelled by the region between the \(x\)-axis and the curve with parametric equations $$x = 1 + 0.2t - \cos t, \quad y = k \sin^2 t,$$ where \(k\) is a positive constant and \(0 \leq t \leq \pi\). Lengths are in metres and the area of the emblem must be \(1 \text{m}^2\).

1. Show that \(k \int_0^\pi (0.2 + \sin t - 0.2 \cos^2 t - \sin t \cos^2 t) dt = 1\). [3]
2. Determine the exact value of \(k\). [6]

At time \(t\) hours after a high tide, the height, \(h\) metres, of the tide and the velocity, \(v\) knots, of the tidal flow can be modelled using the parametric equations $$v = 4 - \left(\frac{2t}{3} - 2\right)^2$$ $$h = 3 - 2\sqrt[3]{t - 3}$$ High tides and low tides occur alternately when the velocity of the tidal flow is zero. A high tide occurs at 2am.

1. 1. Use the model to find the height of this high tide. [1 mark]
   2. Find the time of the first low tide after 2am. [3 marks]
   3. Find the height of this low tide. [1 mark]
2. Use the model to find the height of the tide when it is flowing with maximum velocity. [3 marks]
3. Comment on the validity of the model. [2 marks]

The curve defined by the parametric equations $$x = t^2 \text{ and } y = 2t \quad -\sqrt{2} \leq t \leq \sqrt{2}$$ is shown in Figure 1 below. \includegraphics{figure_1}

1. Find a Cartesian equation of the curve in the form \(y^2 = f(x)\) [2 marks]
2. The point \(A\) lies on the curve where \(t = a\) The tangent to the curve at \(A\) is at an angle \(\theta\) to a line through \(A\) parallel to the \(x\)-axis. The point \(B\) has coordinates \((1, 0)\) The line \(AB\) is at an angle \(\phi\) to the \(x\)-axis. \includegraphics{figure_1_extended}
   1. By considering the gradient of the curve, show that $$\tan \theta = \frac{1}{a}$$ [3 marks]
   2. Find \(\tan \phi\) in terms of \(a\). [2 marks]
   3. Show that \(\tan 2\theta = \tan \phi\) [3 marks]

A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$

1. Find the gradient of the curve at the point where \(t = -2\) [4 marks]
2. Find a Cartesian equation of the curve. [2 marks]

A design for a surfboard is shown in Figure 1. Figure 1 \includegraphics{figure_6_1} The curve of the top half of the surfboard can be modelled by the parametric equations $$x = -2t^2$$ $$y = 9t - 0.7t^2$$ for \(0 \leq t \leq 9.5\) as shown in Figure 2, where \(x\) and \(y\) are measured in centimetres. Figure 2 \includegraphics{figure_6_2}

1. Find the length of the surfboard. [2 marks]
2. 1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [3 marks]
   2. Hence, show that the width of the surfboard is approximately one third of its length. [4 marks]

A water slide is the shape of a curve \(PQ\) as shown in Figure 1 below. \includegraphics{figure_9} The curve can be modelled by the parametric equations $$x = t - \frac{1}{t} + 4.8$$ $$y = t + \frac{2}{t}$$ where \(0.2 \leq t \leq 3\) The horizontal distance from O is \(x\) metres. The vertical distance above the point O at ground level is \(y\) metres. P is the point where \(t = 0.2\) and Q is the point where \(t = 3\)

1. To make sure speeds are safe at Q, the difference in height between P and Q must be less than 7 metres. Show that the slide meets this safety requirement. [3 marks]
2. 1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\) [3 marks]
   2. A vertical support, RS, is to be added between the ground and the lowest point on the slide as shown in Figure 2 below. \includegraphics{figure_9b} Find the length of RS [4 marks]
   3. Find the acute angle the slide makes with the horizontal at Q Give your answer to the nearest degree. [2 marks]

The parametric equations of a curve are \(x = 2 + 5\cos\theta\) and \(y = 1 + 5\sin\theta\), where \(0 \leq \theta < 2\pi\).

1. Determine the cartesian equation of the curve. [3]
2. Hence or otherwise, find the equation of the tangent to the curve at the point \((5, -3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be determined. [4]

The equation of a curve \(C\) is given by the parametric equations $$x = \cos 2\theta, \quad y = \cos\theta.$$

1. Find the Cartesian equation of \(C\). [2]
2. Show that the line \(x - y + 1 = 0\) meets \(C\) at the point \(P\), where \(\theta = \frac{\pi}{3}\), and at the point \(Q\), where \(\theta = \frac{\pi}{2}\). Write down the coordinates of \(P\) and \(Q\). [5]
3. Determine the equations of the tangents to \(C\) at \(P\) and \(Q\). Write down the coordinates of the point of intersection of the two tangents. [7]

The curve \(C_1\) has parametric equations \(x = 3p + 1\), \(y = 9p^2\). The curve \(C_2\) has parametric equations \(x = 4q\), \(y = 2q\). Find the Cartesian coordinates of the points of intersection of \(C_1\) and \(C_2\). [7]

A curve is defined parametrically by $$x = 2\theta + \sin 2\theta, \quad y = 1 + \cos 2\theta.$$

1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(-\tan\theta\). [6]
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{4}\). [4]