1.03g Parametric equations: of curves and conversion to cartesian

8. The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(P\). The point \(B\), which does not lie on the parabola, has coordinates ( \(q , r\) ) where \(q\) and \(r\) are positive constants and \(q > a\). The line \(l\) passes through \(B\) and \(S\).

1. Show that an equation of the line \(l\) is $$( q - a ) y = r ( x - a )$$ The line \(l\) intersects the directrix of \(P\) at the point \(C\). Given that the area of triangle \(O C S\) is three times the area of triangle \(O B S\), where \(O\) is the origin,
2. show that the area of triangle \(O B C\) is \(\frac { 6 } { 5 } \mathrm { qr }\)

8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a t ^ { 2 } , 2 a t \right)\) lies on \(C\).

1. Using calculus, show that the normal to \(C\) at \(P\) has equation $$y + t x = a t ^ { 3 } + 2 a t$$ The point \(S\) is the focus of the parabola \(C\).
   The point \(B\) lies on the positive \(x\)-axis and \(O B = 5 O S\), where \(O\) is the origin.
2. Write down, in terms of \(a\), the coordinates of the point \(B\). A circle has centre \(B\) and touches the parabola \(C\) at two distinct points \(Q\) and \(R\). Given that \(t \neq 0\),
3. find the coordinates of the points \(Q\) and \(R\).
4. Hence find, in terms of \(a\), the area of triangle \(B Q R\).

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)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The rectangular hyperbola \(H\) has equation \(x y = 20\) The point \(P \left( 2 t \sqrt { a } , \frac { 2 \sqrt { a } } { t } \right) , t \neq 0\), where \(a\) is a constant, is a general point on \(H\)

1. State the value of \(a\)
2. Show that the normal to \(H\) at the point \(P\) has equation $$t y - t ^ { 3 } x - 2 \sqrt { 5 } \left( 1 - t ^ { 4 } \right) = 0$$ The points \(A\) and \(B\) lie on \(H\) The point \(A\) has parameter \(t = c\) and the point \(B\) has parameter \(t = - \frac { 1 } { 2 c }\), where \(c\) is a constant. The normal to \(H\) at \(A\) meets \(H\) again at \(B\)
3. Determine the possible values of \(C\)

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

The point \(S\) is the focus of \(C\) The line \(l _ { 1 }\) with equation \(y = k\) where \(k\) is a positive constant, intersects \(C\) at the point \(P\)

1. Show that $$P S = \frac { k ^ { 2 } + 4 a ^ { 2 } } { 4 a }$$ The line \(l _ { 2 }\) passes through \(P\) and intersects the directrix of \(C\) on the \(x\)-axis.
   The line \(l _ { 2 }\) intersects the \(y\)-axis at the point \(A\)
2. Show that the \(y\) coordinate of \(A\) is \(\frac { 4 a ^ { 2 } k } { k ^ { 2 } + 4 a ^ { 2 } }\) The line \(l _ { 1 }\) intersects the directrix of \(C\) at the point \(B\) Given that the areas of triangles \(B P A\) and \(O S P\), where \(O\) is the origin, satisfy the ratio $$\text { area } B P A \text { : area } O S P = 4 k ^ { 2 } : 1$$
3. determine the exact value of \(a\)

1. The hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\).
The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).

1. Determine, in terms of \(c\) and \(t\),
   1. the coordinates of \(A\),
   2. the coordinates of \(B\). Given that the area of triangle \(A O B\), where \(O\) is the origin, is 90 square units,
2. determine the value of \(c\), giving your answer as a simplified surd.

1. The parabola \(C\) has equation \(y ^ { 2 } = \frac { 4 } { 3 } x\)

The point \(P \left( \frac { 1 } { 3 } t ^ { 2 } , \frac { 2 } { 3 } t \right)\), where \(t \neq 0\), lies on \(C\).

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$3 t x + 3 y = t ^ { 3 } + 2 t$$ The normal to \(C\) at the point where \(t = 9\) meets \(C\) again at the point \(Q\).
2. Determine the exact coordinates of \(Q\).

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. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(Q\) lies on the directrix of \(C\). The point \(P\) lies on \(C\) where \(y > 0\) and the line segment \(Q P\) is parallel to the \(x\)-axis. Given that the length of \(P S\) is 13

1. write down the length of \(P Q\). Given that the point \(P\) has \(x\) coordinate 9
   find
2. the value of \(a\),
3. the area of triangle \(P S Q\).

2. A parabola \(P\) has cartesian equation \(y ^ { 2 } = 28 x\). The point \(S\) is the focus of the parabola \(P\).

1. Write down the coordinates of the point \(S\). Points \(A\) and \(B\) lie on the parabola \(P\). The line \(A B\) is parallel to the directrix of \(P\) and cuts the \(x\)-axis at the midpoint of \(O S\), where \(O\) is the origin.
2. Find the exact area of triangle \(A B S\).
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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 \(A B\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 36 x\)

The point \(P \left( 9 t ^ { 2 } , 18 t \right)\), where \(t \neq 0\), lies on \(C\)

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$y + t x = 9 t ^ { 3 } + 18 t$$
2. Hence find the equations of the two normals to \(C\) which pass through the point (54, 0), giving your answers in the form \(y = p x + q\) where \(p\) and \(q\) are constants to be determined. Given that
   • the normals found in part (b) intersect the directrix of \(C\) at the points \(A\) and \(B\)
   • the point \(F\) is the focus of \(C\)
   • determine the area of triangle \(A F B\)

1. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 9\)

The point \(P\) with coordinates \(\left( 3 t , \frac { 3 } { t } \right)\), where \(t \neq 0\), lies on \(H\)

1. Use calculus to determine an equation for the normal to \(H\) at the point \(P\) Give your answer in the form \(t y - t ^ { 3 } x = \mathrm { f } ( t )\) Given that \(t = 2\)
2. determine the coordinates of the point where the normal meets \(H\) again. Give your answer in simplest form.

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\) The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)

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

The point \(P\) on \(C\) has coordinates ( \(5 p ^ { 2 } , 10 p\) ) where \(p\) is a non-zero constant.

1. Use calculus to show that the tangent to \(C\) at \(P\) has equation $$p y - x = 5 p ^ { 2 }$$ The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
2. Write down the coordinates of \(A\). The point \(S\) is the focus of \(C\).
3. Write down the coordinates of \(S\). The straight line \(l _ { 1 }\) passes through \(A\) and \(S\).
   The straight line \(l _ { 2 }\) passes through \(O\) and \(P\), where \(O\) is the origin. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\),
4. show that the coordinates of \(B\) satisfy the equation $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 8 x\).
The point \(P\) lies on \(C\), where \(y > 0\), and the point \(Q\) lies on \(C\), where \(y < 0\) The line segment \(P Q\) is parallel to the \(y\)-axis. Given that the distance \(P Q\) is 12 ,

1. write down the \(y\) coordinate of \(P\),
2. find the \(x\) coordinate of \(P\). Figure 1 shows the point \(S\) which is the focus of \(C\). The line \(l\) passes through the point \(P\) and the point \(S\).
3. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 }$$ This tangent meets the \(y\)-axis at the point \(R\).
2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
3. Show that \(l\) passes through the focus of the parabola.
4. Find the coordinates of the point where \(I\) meets the directrix of the parabola.

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 \(A B\).

8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 } .$$ This tangent meets the \(y\)-axis at the point \(R\).
2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
3. Show that \(l\) passes through the focus of the parabola.
4. Find the coordinates of the point where \(l\) meets the directrix of the parabola.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the parabola with equation \(y ^ { 2 } = 12 x\).
The point \(P\) on the parabola has \(x\)-coordinate \(\frac { 1 } { 3 }\).
The point \(S\) is the focus of the parabola.

1. Write down the coordinates of \(S\). The points \(A\) and \(B\) lie on the directrix of the parabola.
   The point \(A\) is on the \(x\)-axis and the \(y\)-coordinate of \(B\) is positive. Given that \(A B P S\) is a trapezium,
2. calculate the perimeter of \(A B P S\).

7. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a constant. The point \(P \left( c t , \frac { c } { t } \right)\) is a general point on \(H\).

1. Show that the tangent to \(H\) at \(P\) has equation $$t ^ { 2 } y + x = 2 c t$$ The tangents to \(H\) at the points \(A\) and \(B\) meet at the point \(( 15 c , - c )\).
2. Find, in terms of \(c\), the coordinates of \(A\) and \(B\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 36 x\). The point \(S\) is the focus of \(C\).

1. Find the coordinates of \(S\).
2. Write down the equation of the directrix of \(C\). Figure 1 shows the point \(P\) which lies on \(C\), where \(y > 0\), and the point \(Q\) which lies on the directrix of \(C\). The line segment \(Q P\) is parallel to the \(x\)-axis. Given that the distance \(P S\) is 25 ,
3. write down the distance \(Q P\),
4. find the coordinates of \(P\),
5. find the area of the trapezium \(O S P Q\).

10. The point \(P \left( 6 t , \frac { 6 } { t } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\).

1. Show that an equation for the tangent to \(H\) at \(P\) is $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
2. Find the coordinates of \(A\) and \(B\).