Find intersection points

1. The hyperbola \(H\) has Cartesian equation \(x y = 25\)

The parabola \(P\) has parametric equations \(x = 10 t ^ { 2 } , y = 20 t\) The hyperbola \(H\) intersects the parabola \(P\) at the point \(A\)

1. Use algebra to determine the coordinates of \(A\) The point \(B\) with coordinates \(( 10,20 )\) lies on \(P\)
2. Find an equation for the normal to \(P\) at \(B\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined.
3. Use algebra to determine, in simplest form, the exact coordinates of the points where this normal intersects the hyperbola \(H\) (6)

8. The hyperbola \(H\) has cartesian equation \(x y = 16\) The parabola \(P\) has parametric equations \(x = 8 t ^ { 2 } , y = 16 t\).

1. Find, using algebra, the coordinates of the point \(A\) where \(H\) meets \(P\). Another point \(B ( 8,2 )\) lies on the hyperbola \(H\).
2. Find the equation of the normal to \(H\) at the point (8, 2), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
3. Find the coordinates of the points where this normal at \(B\) meets the parabola \(P\).

4. The rectangular hyperbola \(H\) has parametric equations $$x = 4 t , \quad y = \frac { 4 } { t }$$ The straight line with equation \(3 y - 2 x = 10\) intersects \(H\) at the points \(A\) and \(B\). Given that the point \(A\) is above the \(x\)-axis,

1. find the coordinates of the point \(A\) and the coordinates of the point \(B\).
2. Find the coordinates of the midpoint of \(A B\).

3. A rectangular hyperbola has parametric equations $$x = 2 t , \quad y = \frac { 2 } { t } , \quad t \neq 0$$ Points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 2 }\) and \(t = 4\) respectively.
The line \(L\), which passes through the origin \(O\), is perpendicular to the chord \(P Q\).
Find an equation for \(L\).

3. The rectangular hyperbola \(H\) has parametric equations $$x = 4 t , \quad y = \frac { 4 } { t } \quad t \neq 0$$ The points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 4 }\) and \(t = 2\) respectively.
The line \(l\) passes through the origin \(O\) and is perpendicular to the line \(P Q\).

1. Find an equation for \(l\).
2. Find a cartesian equation for \(H\).
3. Find the exact coordinates of the two points where \(l\) intersects \(H\). Give your answers in their simplest form.

3. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) given by the parametric equations $$x = \frac { 5 } { \sqrt { 3 } } \sin t \quad y = 5 ( 1 - \cos t ) \quad 0 \leqslant t \leqslant 2 \pi$$ The circle with centre at the origin \(O\) and with radius \(\frac { 5 \sqrt { 2 } } { 2 }\) meets the curve \(C\) at the points \(A\) and \(B\) as shown in Figure 1.

1. Determine the value of \(t\) at the point \(B\) . The region \(R\) ,shown shaded in Figure 1,is bounded by the curve \(C\) and the circle.
2. Determine the area of the region \(R\) .

12 A curve \(C\) is defined parametrically by $$x = \cos t ( 1 - 2 \sin t ) , \quad y = \sin t ( 1 - 3 \sin t ) , \quad 0 \leqslant t < 2 \pi$$

1. Show that \(C\) intersects the \(y\)-axis at exactly three points, and state the values of \(t\) and \(y\) at these points.
2. Find the range of values of \(t\) for which \(C\) lies above the \(x\)-axis.

The parametric equations of the curve \(C\) are $$x = 3 - 4 t + t ^ { 2 } , \quad y = ( 4 - t ) ^ { 2 }$$ a) Find the coordinates of the points where \(C\) meets the \(y\)-axis.
b) Show that the \(x\)-axis is a tangent to the curve \(C\). a) Prove that $$\cos ( \alpha - \beta ) + \sin ( \alpha + \beta ) \equiv ( \cos \alpha + \sin \alpha ) ( \cos \beta + \sin \beta )$$ b) i) Hence show that \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta }\) can be expressed as \(\cos \theta + \sin \theta\).
ii) Explain why \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta } \neq \cos \theta + \sin \theta\) when \(\theta = \frac { 3 \pi } { 16 }\).

The curve \(C_1\) has parametric equations \(x = 3p + 1\), \(y = 9p^2\). The curve \(C_2\) has parametric equations \(x = 4q\), \(y = 2q\). Find the Cartesian coordinates of the points of intersection of \(C_1\) and \(C_2\). [7]