Parametric point verification

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

1. Write the cartesian equation of \(H\) in the form \(x y = c ^ { 2 }\). Points \(A\) and \(B\) on the hyperbola have parameters \(t = 1\) and \(t = 5\) respectively.
2. Find the coordinates of the mid-point of \(A B\).

5. The parabola \(C\) has equation \(y ^ { 2 } = 20 x\).

1. Verify that the point \(P \left( 5 t ^ { 2 } , 10 t \right)\) is a general point on \(C\). The point \(A\) on \(C\) has parameter \(t = 4\).
   The line \(l\) passes through \(A\) and also passes through the focus of \(C\).
2. Find the gradient of \(l\).

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

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

7 Given that \(x = 2 \sec \theta\) and \(y = 3 \tan \theta\), show that \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1\).

4. \begin{figure}[h] \end{figure} The curve \(C _ { 1 }\) with parametric equations $$x = 10 \cos t , \quad y = 4 \sqrt { 2 } \sin t , \quad 0 \leqslant t < 2 \pi$$ meets the circle \(C _ { 2 }\) with equation $$x ^ { 2 } + y ^ { 2 } = 66$$ at four distinct points as shown in Figure 2.
Given that one of these points, \(S\), lies in the 4th quadrant, find the Cartesian coordinates of \(S\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\) where \(a\) is a positive constant.

The point \(P \left( a t ^ { 2 } , 2 a t \right) , t \neq 0\), lies on \(C\)
The normal to \(C\) at \(P\) is parallel to the line with equation \(y = 2 x\)

1. For the point \(P\), show that \(t = - 2\) The normal to \(C\) at \(P\) intersects \(C\) again when \(x = 9\)
2. Determine the value of \(a\), giving a reason for your answer.

4 Given that \(x = 2 \sec \theta\) and \(y = 3 \tan \theta\), show that \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1\).

8 The diagram shows a part of the curve $$\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 6 } = 1$$ and a chord \(P Q\) of the curve, where \(P\) lies on the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-05_751_1072_680_459}

1. Write down the coordinates of \(P\).
2. The gradient of the chord \(P Q\) is 2 . Find the coordinates of \(Q\).