1.03g Parametric equations: of curves and conversion to cartesian

3. A parabola \(C\) has cartesian equation \(y ^ { 2 } = 16 x\). The point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).

1. Write down the coordinates of the focus \(F\) and the equation of the directrix of \(C\).
2. Show that the equation of the normal to \(C\) at \(P\) is \(y + t x = 8 t + 4 t ^ { 3 }\).

9. The rectangular hyperbola \(H\) has cartesian equation \(x y = 9\) The points \(P \left( 3 p , \frac { 3 } { p } \right)\) and \(Q \left( 3 q , \frac { 3 } { q } \right)\) lie on \(H\), where \(p \neq \pm q\).

1. Show that the equation of the tangent at \(P\) is \(x + p ^ { 2 } y = 6 p\).
2. Write down the equation of the tangent at \(Q\). The tangent at the point \(P\) and the tangent at the point \(Q\) intersect at \(R\).
3. Find, as single fractions in their simplest form, the coordinates of \(R\) in terms of \(p\) and \(q\).

7. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\) The point \(P \left( 5 p , \frac { 5 } { p } \right)\), and the point \(Q \left( 5 q , \frac { 5 } { q } \right)\), where \(p , q \neq 0 , p \neq q\), are points on the rectangular hyperbola \(H\).

1. Show that the equation of the tangent at point \(P\) is $$p ^ { 2 } y + x = 10 p$$
2. Write down the equation of the tangent at point \(Q\). The tangents at \(P\) and \(Q\) meet at the point \(N\).
   Given \(p + q \neq 0\),
3. show that point \(N\) has coordinates \(\left( \frac { 10 p q } { p + q } , \frac { 10 } { p + q } \right)\). The line joining \(N\) to the origin is perpendicular to the line \(P Q\).
4. Find the value of \(p ^ { 2 } q ^ { 2 }\).

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the parabola with equation \(y ^ { 2 } = 36 x\).
The point \(P ( 4,12 )\) lies on the parabola.

1. Find an equation for the normal to the parabola at \(P\). This normal meets the \(x\)-axis at the point \(N\) and \(S\) is the focus of the parabola, as shown in Figure 1.
2. Find the area of triangle \(P S N\).

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 \(L\).

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

1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = a p ^ { 3 } + 2 a p$$ The normal to \(C\) at the point \(P\) meets the \(x\)-axis at the point \(( 6 a , 0 )\) and meets the directrix of \(C\) at the point \(D\). Given that \(p > 0\),
2. find, in terms of \(a\), the coordinates of \(D\). Given also that the directrix of \(C\) cuts the \(x\)-axis at the point \(X\),
3. find, in terms of \(a\), the area of the triangle XPD, giving your answer in its simplest form.

6. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\).

1. Verify that the point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
2. Write down the coordinates of the focus \(S\) of \(C\).
3. Show that the normal to \(C\) at \(P\) has equation $$y + t x = 8 t + 4 t ^ { 3 }$$ The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(N\).
4. Find the area of triangle \(P S N\) in terms of \(t\), giving your answer in its simplest form.

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 \(l\).

8. The parabola \(C\) has equation \(y ^ { 2 } = 48 x\). The point \(P \left( 12 t ^ { 2 } , 24 t \right)\) is a general point on \(C\).

1. Find the equation of the directrix of \(C\).
2. Show that the equation of the tangent to \(C\) at \(P \left( 12 t ^ { 2 } , 24 t \right)\) is $$x - t y + 12 t ^ { 2 } = 0$$ The tangent to \(C\) at the point \(( 3,12 )\) meets the directrix of \(C\) at the point \(X\).
3. Find the coordinates of \(X\).

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

1. Show that an equation for the tangent to \(H\) at \(P\) is $$x + t ^ { 2 } y = 2 c t$$ The tangent to \(H\) at the point \(P\) meets the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Given that the area of the triangle \(O A B\), where \(O\) is the origin, is 36 ,
2. find the exact value of \(c\), expressing your answer in the form \(k \sqrt { } 2\), where \(k\) is an integer.

6. A curve \(C\) is in the form of a parabola with equation \(y ^ { 2 } = 4 x\). \(P \left( p ^ { 2 } , 2 p \right)\) and \(Q \left( q ^ { 2 } , 2 q \right)\) are points on \(C\) where \(p > q\).

1. Find an equation of the tangent to \(C\) at \(P\).
   (5)
2. The tangent at \(P\) and the tangent at \(Q\) are perpendicular and intersect at the point \(R ( - 1,2 )\).
   1. Find the exact value of \(p\) and the exact value of \(q\).
   2. Find the area of the triangle \(P Q R\).

4. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 4\) The point \(P \left( 2 t , \frac { 2 } { t } \right)\) lies on \(H\), where \(t \neq 0\)

1. Show that an equation of the normal to \(H\) at the point \(P\) is $$t y - t ^ { 3 } x = 2 - 2 t ^ { 4 }$$ The normal to \(H\) at the point where \(t = - \frac { 1 } { 2 }\) meets \(H\) again at the point \(Q\).
2. Find the coordinates of the point \(Q\).

6. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x , \quad a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0 , p \neq q\).

1. Show that an equation of the tangent to the parabola at \(P\) is $$p y - x = a p ^ { 2 }$$
2. Write down the equation of the tangent at \(Q\). The tangent at \(P\) meets the tangent at \(Q\) at the point \(R\).
3. Find, in terms of \(p\) and \(q\), the coordinates of \(R\), giving your answers in their simplest form. Given that \(R\) lies on the directrix of \(C\),
4. find the value of \(p q\).

7. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 4 a x , a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(P ^ { \prime } \left( a p ^ { 2 } , - 2 a p \right)\) lie on \(C\).

1. Show that an equation of the normal to \(C\) at the point \(P\) is $$y + p x = 2 a p + a p ^ { 3 }$$
2. Write down an equation of the normal to \(C\) at the point \(P ^ { \prime }\). The normal to \(C\) at \(P\) meets the normal to \(C\) at \(P ^ { \prime }\) at the point \(Q\).
3. Find, in terms of \(a\) and \(p\), the coordinates of \(Q\). Given that \(S\) is the focus of the parabola,
4. find the area of the quadrilateral \(S P Q P ^ { \prime }\).

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\). An equation for the tangent to \(H\) at \(P\) is given by $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 2 c } { t }$$ The points \(A\) and \(B\) lie on \(H\).
The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 6 } { 7 } c , \frac { 12 } { 7 } c \right)\).
Find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).

6. The rectangular hyperbola \(H\) has cartesian equation \(x y = c ^ { 2 }\). The point \(P \left( c t , \frac { c } { t } \right) , t > 0\), is a general point on \(H\).

1. Show that an equation of the tangent to \(H\) at the point \(P\) is $$t ^ { 2 } y + x = 2 c t$$ An equation of the normal to \(H\) at the point \(P\) is \(t ^ { 3 } x - t y = c t ^ { 4 } - c\) Given that the normal to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and the tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(B\),
2. find, in terms of \(c\) and \(t\), the coordinates of \(A\) and the coordinates of \(B\). Given that \(c = 4\),
3. find, in terms of \(t\), the area of the triangle \(A P B\). Give your answer in its simplest form.

8. The points \(P \left( 4 k ^ { 2 } , 8 k \right)\) and \(Q \left( k ^ { 2 } , 4 k \right)\), where \(k\) is a constant, lie on the parabola \(C\) with equation \(y ^ { 2 } = 16 x\). The straight line \(l _ { 1 }\) passes through the points \(P\) and \(Q\).

1. Show that an equation of the line \(l _ { 1 }\) is given by $$3 k y - 4 x = 8 k ^ { 2 }$$ The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the focus of the parabola \(C\). The line \(l _ { 2 }\) meets the directrix of \(C\) at the point \(R\).
2. Find, in terms of \(k\), the \(y\) coordinate of the point \(R\).

5. Points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p ^ { 2 } \neq q ^ { 2 }\), lie on the parabola \(y ^ { 2 } = 4 a x\).

1. Show that the chord \(P Q\) has equation $$y ( p + q ) = 2 x + 2 a p q$$ Given that this chord passes through the focus of the parabola,
2. show that \(p q = - 1\)
3. Using calculus find the gradient of the tangent to the parabola at \(P\).
4. Show that the tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) are perpendicular.
   

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.

7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant and \(a > 0\) The point \(Q \left( a q ^ { 2 } , 2 a q \right) , q > 0\), lies on the parabola \(C\).

1. Show that an equation of the tangent to \(C\) at \(Q\) is $$q y = x + a q ^ { 2 }$$ The tangent to \(C\) at the point \(Q\) meets the \(x\)-axis at the point \(X \left( - \frac { 1 } { 4 } a , 0 \right)\) and meets the directrix of \(C\) at the point \(D\).
2. Find, in terms of \(a\), the coordinates of \(D\). Given that the point \(F\) is the focus of the parabola \(C\),
3. find the area, in terms of \(a\), of the triangle \(F X D\), giving your answer in its simplest form.

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

Given that \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\),

1. use calculus to show that the equation of the tangent to \(H\) at \(P\) can be written as $$t ^ { 2 } y + x = 2 c t$$ The points \(A\) and \(B\) lie on \(H\).
   The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 8 c } { 5 } , \frac { 3 c } { 5 } \right)\).
   Given that the \(x\) coordinate of \(A\) is positive,
2. find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\). The point ( \(3 t , \frac { 3 } { t }\) ) is a general point on this hyperbola.

1. Find the value of \(c ^ { 2 }\).
2. Show that an equation of the normal to \(H\) at the point ( \(3 t , \frac { 3 } { t }\) ) is $$y = t ^ { 2 } x + \left( \frac { 3 } { t } - 3 t ^ { 3 } \right)$$ The point \(P\) on \(H\) has coordinates (6, 1.5). The tangent to \(H\) at \(P\) meets the curve again at the point \(Q\).
3. Find the coordinates of the point \(Q\).

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

1. Find the value of \(a\).
2. Show that the equation for the tangent to \(C\) at the point \(\left( 4 t ^ { 2 } , 8 t \right)\) is $$y t = x + 4 t ^ { 2 } .$$ The tangent to \(C\) at the point \(A\) meets the tangent to \(C\) at the point \(B\) on the directrix of \(C\) when \(y = 15\).
3. Find the coordinates of \(A\) and the coordinates of \(B\).