1.03g Parametric equations: of curves and conversion to cartesian

The point \(P(ap^2, 2ap)\) lies on the parabola \(M\) with equation \(y^2 = 4ax\), where \(a\) is a positive constant.

1. Show that an equation of the tangent to \(M\) at \(P\) is \(py = x + ap^2\). [3]

The point \(Q(16ap^2, 8ap)\) also lies on \(M\).

1. Write down an equation of the tangent to \(M\) at \(Q\). [2]

The parabola \(C\) has equation \(y^2 = 4ax\), where \(a\) is a constant.

1. Show that an equation for the normal to \(C\) at the point \(P(ap^2, 2ap)\) is \(y + px = 2ap + ap^3\). [4]

The normals to \(C\) at the points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), \(p \neq q\), meet at the point \(R\).

1. Find, in terms of \(a\), \(p\) and \(q\), the coordinates of \(R\). [5]

The points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), \(p \neq q\), lie on the parabola \(C\) with equation \(y^2 = 4ax\), where \(a\) is a constant.

1. Show that an equation for the chord \(PQ\) is \((p + q) y = 2(x + apq)\) . [3]

The normals to \(C\) at \(P\) and \(Q\) meet at the point \(R\).

1. Show that the coordinates of \(R\) are \((a(p^2 + q^2 + pq + 2), -apq(p + q) )\). [7]

The hyperbola \(H\) has equation $$4x^2 - y^2 = 4$$

1. Write down the equations of the asymptotes of \(H\). [1]
2. Find the coordinates of the foci of \(H\). [2]

The point \(P(\sec \theta, 2 \tan \theta)\) lies on \(H\).

1. Using calculus, show that the equation of the tangent to \(H\) at the point \(P\) is $$y \tan \theta = 2x \sec \theta - 2$$ [4]

The point \(V(-1, 0)\) and the point \(W(1, 0)\) both lie on \(H\). The point \(Q(\sec \theta, -2 \tan \theta)\) also lies on \(H\). Given that \(P\), \(Q\), \(V\) and \(W\) are distinct points on \(H\) and that the lines \(VP\) and \(WQ\) intersect at the point \(S\),

1. show that, as \(\theta\) varies, \(S\) lies on an ellipse with equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where \(a\) and \(b\) are integers to be found. [7]

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

1. Show that an equation for \(L\) is $$5x \sin \theta - 2y \cos \theta = 21 \sin \theta \cos \theta$$ [5]

Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(PQ\),

1. find the exact area of triangle \(OPM\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2\theta\) [6]

The curve \(C\) has parametric equations $$x = 3t^4, \quad y = 4t^3, \quad 0 \leq t \leq 1$$ The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).

1. Show that $$S = k\pi \int_{0}^{1} t^2(t^2 + 1)^{\frac{1}{2}} dt$$ where \(k\) is a constant to be found. [4]
2. Use the substitution \(u^2 = t^2 + 1\) to find the value of \(S\), giving your answer in the form \(p\pi\left(11\sqrt{2} - 4\right)\) where \(p\) is a rational number to be found. [7]

[In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.] The points \(P(3\cos \alpha, 2\sin \alpha)\) and \(Q(3\cos \beta, 2\sin \beta)\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$

1. Show the equation of the chord \(PQ\) is $$\frac{x}{3}\cos\frac{(\alpha + \beta)}{2} + \frac{y}{2}\sin\frac{(\alpha + \beta)}{2} = \cos\frac{(\alpha - \beta)}{2}$$ [4]
2. Write down the coordinates of the mid-point of \(PQ\). [1]

Given that the gradient, \(m\), of the chord \(PQ\) is a constant,

1. show that the centre of the chord lies on a line $$y = -kx$$ expressing \(k\) in terms of \(m\). [5]

\includegraphics{figure_25} Figure 1 shows the curve with parametric equations $$x = a \cos^3 \theta, \quad y = a \sin^3 \theta, \quad 0 \leq \theta < 2\pi.$$

1. Find the total length of this curve. [7]

The curve is rotated through \(\pi\) radians about the \(x\)-axis.

1. Find the area of the surface generated. [5]

\includegraphics{figure_31} Figure 1 shows a sketch of the curve with parametric equations $$x = a \cos^3 t, \quad y = a \sin^3 t, \quad 0 \leq t \leq \frac{\pi}{2},$$ where \(a\) is a positive constant. The curve is rotated through \(2\pi\) radians about the \(x\)-axis. Find the exact value of the area of the curved surface generated. [8]

\includegraphics{figure_1} The parametric equations of the curve \(C\) shown in Figure 1 are $$x = a(t - \sin t), \quad y = a(1 - \cos t), \quad 0 \leq t \leq 2\pi$$ Find, by using integration, the length of \(C\). (Total 6 marks)

The hyperbola \(C\) has equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

1. Show that an equation of the normal to \(C\) at \(P(a \sec \theta, b \tan \theta)\) is $$by + ax \sin \theta = (a^2 + b^2)\tan \theta$$ [6] The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(AB\) is \(M\).
2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies. [7]

(Total 13 marks)

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

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3 marks]
2. Find an equation of the normal to the curve at the point where \(t = 1\). [4 marks]
3. Find a cartesian equation of the curve. [2 marks]

A curve is given parametrically by the equations $$x = 5 \cos t, \quad y = -2 + 4 \sin t, \quad 0 \leq t < 2\pi.$$

1. Find the coordinates of all the points at which \(C\) intersects the coordinate axes, giving your answers in surd form where appropriate. [4]
2. Sketch the graph at \(C\). [2]

\(P\) is the point on \(C\) where \(t = \frac{1}{6}\pi\).

1. Show that the normal to \(C\) at \(P\) has equation $$8\sqrt{3}y = 10x - 25\sqrt{3}.$$ [4]

\includegraphics{figure_1} Figure 1 shows a cross-section \(R\) of a dam. The line \(AC\) is the vertical face of the dam, \(AB\) is the horizontal base and the curve \(BC\) is the profile. Taking \(x\) and \(y\) to be the horizontal and vertical axes, then \(A\), \(B\) and \(C\) have coordinates \((0, 0)\), \((3\pi^2, 0)\) and \((0, 30)\) respectively. The area of the cross-section is to be calculated. Initially the profile \(BC\) is approximated by a straight line.

1. Find an estimate for the area of the cross-section \(R\) using this approximation. [1]

The profile \(BC\) is actually described by the parametric equations. $$x = 16t^2 - \pi^2, \quad y = 30 \sin 2t, \quad \frac{\pi}{4} \leq t \leq \frac{\pi}{2}.$$

1. Find the exact area of the cross-section \(R\). [7]
2. Calculate the percentage error in the estimate of the area of the cross-section \(R\) that you found in part (a). [2]

The parametric equations of a curve are \(x = 2t^2\), \(y = 4t\). Two points on the curve are \(P(2p^2, 4p)\) and \(Q(2q^2, 4q)\).

1. Show that the gradient of the normal to the curve at \(P\) is \(-p\). [2]
2. Show that the gradient of the chord joining the points \(P\) and \(Q\) is \(\frac{2}{p + q}\). [2]
3. The chord \(PQ\) is the normal to the curve at \(P\). Show that \(p^2 + pq + 2 = 0\). [2]
4. The normal at the point \(R(8, 8)\) meets the curve again at \(S\). The normal at \(S\) meets the curve again at \(T\). Find the coordinates of \(T\). [4]

A curve is given parametrically by the equations $$x = t^2, \quad y = \frac{1}{t}.$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\), giving your answer in its simplest form. [3]
2. Show that the equation of the tangent at the point \(P\left(4, -\frac{1}{4}\right)\) is \(x - 16y = 12\). [3]
3. Find the value of the parameter at the point where the tangent at \(P\) meets the curve again. [4]

A curve is given parametrically by the equations $$x = 4\cos t, \quad y = 3\sin t,$$ where \(0 \leq t \leq \frac{1}{2}\pi\).

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
2. Show that the equation of the tangent at the point \(P\), where \(t = p\), is $$3x\cos p + 4y\sin p = 12.$$ [3]
3. The tangent at \(P\) meets the \(x\)-axis at \(R\) and the \(y\)-axis at \(S\). \(O\) is the origin. Show that the area of triangle \(ORS\) is \(\frac{6}{\sin 2p}\). [3]
4. Write down the least possible value of the area of triangle \(ORS\), and give the corresponding value of \(p\). [3]

Fig. 8 shows a cross-section of a car headlight whose inside reflective surface is modelled, in suitable units, by the curve $$x = 2t^2, y = 4t, \quad -\sqrt{2} < t < \sqrt{2}.$$ P\((2t^2, 4t)\) is a point on the curve with parameter \(t\). TS is the tangent to the curve at P, and PR is the line through P parallel to the \(x\)-axis. Q is the point (2, 0). The angles that PS and QP make with the positive \(x\)-direction are \(\theta\) and \(\phi\) respectively. \includegraphics{figure_8}

1. By considering the gradient of the tangent TS, show that \(\tan \theta = \frac{1}{t}\). [3]
2. Find the gradient of the line QP in terms of \(t\). Hence show that \(\phi = 2\theta\), and that angle TPQ is equal to \(\theta\). [8]

[The above result shows that if a lamp bulb is placed at Q, then the light from the bulb is reflected to produce a parallel beam of light.] The inside surface of the headlight has the shape produced by rotating the curve about the \(x\)-axis.

1. Show that the curve has cartesian equation \(y^2 = 8x\). Hence find the volume of revolution of the curve, giving your answer as a multiple of \(\pi\). [7]

A curve has parametric equations $$x = at^3, \quad y = \frac{a}{1+t^2},$$ where \(a\) is a constant. Show that \(\frac{dy}{dx} = \frac{-2}{3t(1+t^2)^2}\). Hence find the gradient of the curve at the point \((a, \frac{1}{2}a)\). [7]

A curve has parametric equations $$x = 2 \sin \theta, \quad y = \cos 2\theta.$$

1. Find the exact coordinates and the gradient of the curve at the point with parameter \(\theta = \frac{1}{4}\pi\). [5]
2. Find \(y\) in terms of \(x\). [2]

Fig. 7a shows the curve with the parametric equations $$x = 2\cos\theta, \quad y = \sin 2\theta, \quad -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}.$$ The curve meets the \(x\)-axis at O and P. Q and R are turning points on the curve. The scales on the axes are the same. \includegraphics{figure_7a}

1. State, with their coordinates, the points on the curve for which \(\theta = -\frac{\pi}{2}\), \(\theta = 0\) and \(\theta = \frac{\pi}{2}\). [3]
2. Find \(\frac{dy}{dx}\) in terms of \(\theta\). Hence find the gradient of the curve when \(\theta = \frac{\pi}{2}\), and verify that the two tangents to the curve at the origin meet at right angles. [5]
3. Find the exact coordinates of the turning point Q. [3]

When the curve is rotated about the \(x\)-axis, it forms a paperweight shape, as shown in Fig. 7b. \includegraphics{figure_7b}

1. Express \(\sin^2\theta\) in terms of \(x\). Hence show that the cartesian equation of the curve is \(y^2 = x^2(1 - \frac{1}{4}x^2)\). [4]
2. Find the volume of the paperweight shape. [4]

Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u, \quad y = u + \frac{1}{u}, \quad 1 \leq u \leq 10.$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \includegraphics{figure_7}

1. Find the lengths OA, OB and AC. [5]
2. Find \(\frac{dy}{dx}\) in terms of \(u\). Hence find the angle \(\theta\). [6]
3. Show that the cartesian equation of the curve is \(y = e^{x/5} + e^{-x/5}\). [2]

An object is formed by rotating the region OACB through \(360°\) about Ox.

1. Find the volume of the object. [5]

A curve has parametric equations \(x = e^{2t}, y = te^{2t}\).

1. Find \(\frac{dy}{dx}\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\). [4]
2. Find the cartesian equation of the curve in the form \(y = ax^b \ln x\), where \(a\) and \(b\) are constants to be determined. [3]

A curve has parametric equations $$x = 3 \cos^2 t, \quad y = \sin 2t, \quad 0 \leq t < \pi.$$

1. Show that \(\frac{dy}{dx} = -\frac{2}{3} \cot 2t\). [4]
2. Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis. [3]
3. Show that the tangent to the curve at the point where \(t = \frac{\pi}{6}\) has the equation $$2x + 3\sqrt{3} y = 9.$$ [3]
4. Find a cartesian equation for the curve in the form \(y^2 = \text{f}(x)\). [4]

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

1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\left(\frac{2-t}{1+t}\right)^2\). [4]
2. Find an equation for the normal to the curve at the point where \(t = 1\). [3]
3. Show that the cartesian equation of the curve can be written in the form $$y = \frac{1+x}{1+3x}.$$ [4]