Find normal equation

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 \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\) where \(a , b\) and \(c\) are integers.

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.
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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.

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 \(P\).

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

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

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.

8.
\includegraphics[max width=\textwidth, alt={}, center]{85427816-dcf1-49af-8d68-f4e88fc7d8f1-3_497_784_246_461} The diagram shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point \(P\) on the curve has coordinates \(( 1 , \sqrt { 6 } )\).

1. Find the value of \(\theta\) at \(P\).
2. Show that the normal to the curve at \(P\) passes through the origin.
3. Find a cartesian equation for the curve.