Find normal equation at parameter

4 A curve is defined by the parametric equations $$x = 4 \cos ^ { 2 } t , \quad y = \sqrt { 3 } \sin 2 t ,$$ for values of \(t\) such that \(0 < t < \frac { 1 } { 2 } \pi\) .
Find the equation of the normal to the curve at the point for which \(t = \frac { 1 } { 6 } \pi\) .Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-06_2718_35_141_2012}

6 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t } .$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }\).
2. Find the equation of the normal to the curve at the point where \(t = 0\).

5 The parametric equations of a curve are $$x = \ln ( t + 1 ) , \quad y = \mathrm { e } ^ { 2 t } + 2 t$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the normal to the curve at the point for which \(t = 0\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t }$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }\).
2. Find the equation of the normal to the curve at the point where \(t = 0\).

7 The parametric equations of a curve are $$x = t ^ { 3 } + 6 t + 1 , \quad y = t ^ { 4 } - 2 t ^ { 3 } + 4 t ^ { 2 } - 12 t + 5$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and use division to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be written in the form \(a t + b\), where \(a\) and \(b\) are constants to be found.
2. The straight line \(x - 2 y + 9 = 0\) is the normal to the curve at the point \(P\). Find the coordinates of \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-3_533_698_735_717} The diagram shows the curve with parametric equations $$x = 4 \sin \theta , \quad y = 1 + 3 \cos \left( \theta + \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant \theta < 2 \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(k ( 1 + ( \sqrt { } 3 ) \tan \theta )\) where the exact value of \(k\) is to be determined.
2. Find the equation of the normal to the curve at the point where the curve crosses the positive \(y\)-axis. Give your answer in the form \(y = m x + c\), where the constants \(m\) and \(c\) are exact.

7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.

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 \(k\).

1. A curve has parametric equations

$$x = 7 \cos t - \cos 7 t , y = 7 \sin t - \sin 7 t , \quad \frac { \pi } { 8 } < t < \frac { \pi } { 3 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). You need not simplify your answer.
2. Find an equation of the normal to the curve at the point where \(t = \frac { \pi } { 6 }\). Give your answer in its simplest exact form.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$

1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
2. Find an equation of the normal to the curve at \(P\).
3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).

4

1. A curve is defined by the parametric equations \(x = 3 \cos 2 \theta , y = 2 \cos \theta\).
   1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }\), where \(k\) is an integer.
   2. Find an equation of the normal to the curve at the point where \(\theta = \frac { \pi } { 3 }\).
2. Find the exact value of \(\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x\).

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The point \(P\) lies on \(C\) and has coordinates \((3, 2)\). The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).

1. [(a)] Find the \(x\) coordinate of the point \(Q\). \hfill [6]

The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.

1. [(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p\pi + q\pi^2\), where \(p\) and \(q\) are rational numbers to be determined. [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}

A curve is defined by the parametric equations $$x = 4\cos^2 t, \quad y = \sqrt{3}\sin 2t,$$ for values of \(t\) such that \(0 < t < \frac{1}{2}\pi\). Find the equation of the normal to the curve at the point for which \(t = \frac{1}{6}\pi\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [7]

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 $$x = t^2 + 1, \quad y = 4t + \ln(2t - 1).$$

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]

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]

The curve defined by the parametric equations $$x = 2\cos\theta, \quad y = 3\sin(2\theta) \quad \text{and} \quad \theta \in [0, 2\pi]$$ is shown below. The point \(P\left(\sqrt{3}, \frac{3\sqrt{3}}{2}\right)\) is marked on the curve. \includegraphics{figure_curve}

1. Show that the equation of the normal to the curve at \(P\) can be written as \(3y - x = \frac{7\sqrt{3}}{2}\) [5]
2. Show that the Cartesian equation of the curve may be written as \(ay^2 + bx^4 + cx^2 = 0\) where \(a\), \(b\) and \(c\) are integers to be found. [3]