1.03g Parametric equations: of curves and conversion to cartesian

A curve has parametric equations $$x = \ln(t + 1), \quad y = t^2 \ln t.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [5]
2. Find the exact value of \(t\) at the stationary point. [2]
3. Find the gradient of the curve at the point where it crosses the \(x\)-axis. [2]

A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$

1. Find the equation of the tangent to the curve at the origin. [5]
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

The parametric equations of a curve are $$x = \tan^2 2t, \quad y = \cos 2t,$$ for \(0 < t < \frac{1}{4}\pi\).

1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\cos^3 2t\). [4]
2. Hence find the equation of the normal to the curve at the point where \(t = \frac{1}{8}\pi\). Give your answer in the form \(y = mx + c\). [4]

The parametric equations of a curve are $$x = 2\theta + \sin 2\theta, \quad y = 1 - \cos 2\theta.$$ Show that \(\frac{dy}{dx} = \tan \theta\). [5]

The parametric equations of a curve are $$x = t - \tan t, \quad y = \ln(\cos t),$$ for \(-\frac{1}{4}\pi < t < \frac{1}{4}\pi\).

1. Show that \(\frac{dy}{dx} = \cot t\). [5]
2. Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2. Give your answer correct to 3 significant figures. [2]

The parametric equations of a curve are $$x = t^2 + 1, \quad y = 4t + \ln(2t - 1).$$

1. Express \(\frac{dy}{dx}\) in terms of \(t\). [3]
2. Find the equation of the normal to the curve at the point where \(t = 1\). Give your answer in the form \(ax + by + c = 0\). [3]

The parametric equations of a curve are $$x = e^{-t}\cos t, \quad y = e^{-t}\sin t.$$ Show that \(\frac{dy}{dx} = \tan(t - \frac{1}{4}\pi)\). [6]

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

1. Obtain an expression for \(\frac{dy}{dx}\) in terms of \(\theta\). [3]
2. Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the \(y\)-axis. [4]

\includegraphics{figure_2} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t^2 + 2t \quad y = \frac{2}{t(3-t)} \quad a \leq t \leq b$$ where \(a\) and \(b\) are constants. The ends of the curve lie on the line with equation \(y = 1\)

1. Find the value of \(a\) and the value of \(b\) [2]

The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)

1. Show that the area of region \(R\) is given by $$M - k \int_a^b \frac{t+1}{t(3-t)} dt$$ where \(M\) and \(k\) are constants to be found. [5]
2. 1. Write \(\frac{t+1}{t(3-t)}\) in partial fractions.
   2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form. [6]

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]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + 3\tan t, \quad y = 2\cos 2t, \quad -\frac{\pi}{6} \leq t \leq \frac{\pi}{3}$$ The curve crosses the \(x\)-axis at point \(P\), as shown in Figure 3.

1. Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be found. [5]

The curve \(C\) has equation \(y = f(x)\), where \(f\) is a function with domain \(\left[k, 1 + 3\sqrt{3}\right]\)

1. Find the exact value of the constant \(k\). [1]
2. Find the range of \(f\). [2]

A curve has parametric equations $$x = 2\cot t, \quad y = 2\sin^2 t, \quad 0 < t \leq \frac{\pi}{2}.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of the parameter \(t\). [4]
2. Find an equation of the tangent to the curve at the point where \(t = \frac{\pi}{4}\). [4]
3. Find a cartesian equation of the curve in the form \(y = f(x)\). State the domain on which the curve is defined. [4]

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

1. Find the value of \(\frac{dy}{dx}\) at the point on \(C\) with coordinates \((9, 5)\). [4]
2. Find a cartesian equation of the curve in the form $$y = \frac{ax + b}{cx + d}$$ where \(a\), \(b\), \(c\) and \(d\) are integers. [3]

A curve \(C\) has parametric equations $$x = 4t + 3, \quad y = 4t + 8 + \frac{5}{2t}, \quad t \neq 0$$

1. Find the value of \(\frac{dy}{dx}\) at the point on \(C\) where \(t = 2\), giving your answer as a fraction in its simplest form. [3]
2. Show that the cartesian equation of the curve \(C\) can be written in the form $$y = \frac{x^2 + ax + b}{x - 3}, \quad x \neq 3$$ where \(a\) and \(b\) are integers to be determined. [3]

\includegraphics{figure_1} The curve shown in Fig. 1 has parametric equations $$x = \cos t, \quad y = \sin 2t, \quad 0 \leq t < 2\pi.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of the parameter \(t\). [3]
2. Find the values of the parameter \(t\) at the points where \(\frac{dy}{dx} = 0\). [3]
3. Hence give the exact values of the coordinates of the points on the curve where the tangents are parallel to the \(x\)-axis. [2]
4. Show that a cartesian equation for the part of the curve where \(0 \leq t < \pi\) is $$y = 2x\sqrt{(1 - x^2)}.$$ [3]
5. Write down a cartesian equation for the part of the curve where \(\pi \leq t < 2\pi\). [1]

\includegraphics{figure_2} Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle \(ABCD\) is red glass. The loops are described by the curve with parametric equations $$x = 3 \cos t, \quad y = 9 \sin 2t, \quad 0 \leq t < 2\pi.$$

1. Find the cartesian equation of the curve in the form \(y^2 = f(x)\). [4]
2. Show that the shaded area in Fig. 2, enclosed by the curve and the \(x\)-axis, is given by $$\int_0^{\frac{\pi}{2}} A \sin 2t \sin t \, dt, \text{ stating the value of the constant } A.$$ [3]
3. Find the value of this integral. [4]

The sides of the rectangle \(ABCD\), in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm,

1. find the total area of the red glass. [4]

\includegraphics{figure_1} A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown in Fig. 1. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a second type of wood. The ellipse has parametric equations, $$x = 5 \cos \theta, \quad y = 4 \sin \theta, \quad 0 \leq \theta < 2\pi$$ The parallelogram consists of four line segments, which are tangents to the ellipse at the points where \(\theta = \alpha\), \(\theta = -\alpha\), \(\theta = \pi - \alpha\), \(\theta = -\pi + \alpha\).

1. Find an equation of the tangent to the ellipse at \((5 \cos \alpha, 4 \sin \alpha)\), and show that it can be written in the form $$5y \sin \alpha + 4x \cos \alpha = 20.$$ [4]
2. Find by integration the area enclosed by the ellipse. [4]
3. Hence show that the area enclosed between the ellipse and the parallelogram is $$\frac{80}{\sin 2\alpha} - 20\pi.$$ [4]
4. Given that \(0 < \alpha < \frac{\pi}{4}\), find the value of \(\alpha\) for which the areas of two types of wood are equal. [3]

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 \(t = \frac{1}{2}\pi\).

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

The parabola \(C\) has equation \(y^2 = 18x\) The point \(S\) is the focus of \(C\)

1. Write down the coordinates of \(S\) [1]

The point \(P\), with \(y > 0\), lies on \(C\) The shortest distance from \(P\) to the directrix of \(C\) is 9 units.

1. Determine the exact perimeter of the triangle \(OPS\), where \(O\) is the origin. Give your answer in simplest form. [4]

The rectangular hyperbola, \(H\), has parametric equations \(x = 5t, y = \frac{5}{t}, t \neq 0\).

1. Write the cartesian equation of \(H\) in the form \(xy = c^2\). [1]
2. Points \(A\) and \(B\) on the hyperbola have parameters \(t = 1\) and \(t = 5\) respectively. Find the coordinates of the mid-point of \(AB\). [3]

A parabola has equation \(y^2 = 4ax\), \(a > 0\). The point \(Q (aq^2, 2aq)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$yq = x + aq^2.$$ [4]
2. This tangent meets the \(y\)-axis at the point \(R\). Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\). [3]
3. Show that \(l\) passes through the focus of the parabola. [1]
4. Find the coordinates of the point where \(l\) meets the directrix of the parabola. [2]

\includegraphics{figure_1} Figure 1 shows a rectangular hyperbola \(H\) with parametric equations $$x = 3t, \quad y = \frac{3}{t}, \quad t \neq 0$$ The line \(L\) with equation \(6y = 4x - 15\) intersects \(H\) at the point \(P\) and at the point \(Q\) as shown in Figure 1.

1. Show that \(L\) intersects \(H\) where \(4t^2 - 5t - 6 = 0\) [3]
2. Hence, or otherwise, find the coordinates of points \(P\) and \(Q\). [5]

The parabola \(C\) has equation \(y^2 = 4ax\), where \(a\) is a positive constant. The point \(P(at^2, 2at)\) is a general point on \(C\).

1. Show that the equation of the tangent to \(C\) at \(P(at^2, 2at)\) is $$ty = x + at^2$$ [4]

The tangent to \(C\) at \(P\) meets the \(y\)-axis at a point \(Q\).

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

Given that the point \(S\) is the focus of \(C\),

1. show that \(PQ\) is perpendicular to \(SQ\). [3]

The point \(P \left( 2p, \frac{2}{p} \right)\) and the point \(Q \left( 2q, \frac{2}{q} \right)\), where \(p \neq -q\), lie on the rectangular hyperbola with equation \(xy = 4\). The tangents to the curve at the points \(P\) and \(Q\) meet at the point \(R\). Show that at the point \(R\), $$x = \frac{4pq}{p + q} \text{ and } y = \frac{4}{p + q}.$$ [8]

Show that the normal to the rectangular hyperbola \(xy = c^2\), at the point \(P \left( ct, \frac{c}{t} \right)\), \(t \neq 0\) has equation $$y = t^2 x + \frac{c}{t} - ct^3.$$ [5]