Normal passes through specific point verification

5 The parametric equations of a curve are $$x = t \mathrm { e } ^ { 2 t } , \quad y = t ^ { 2 } + t + 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 t }\).
2. Hence show that the normal to the curve, where \(t = - 1\), passes through the point \(\left( 0,3 - \frac { 1 } { \mathrm { e } ^ { 4 } } \right)\).

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.

7 A curve is given parametrically by \(x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t\) where \(t\) is any real number.

1. Show that the equation of the normal to the curve at the point where \(t = 2\) can be written in the form \(2 x + 5 y = 30\).
2. Show that this normal does not meet the curve again.