1.07s Parametric and implicit differentiation

6. The curve \(H\) has equation $$x y = a ^ { 2 } \quad x > 0$$ where \(a\) is a positive constant. The line with equation \(y = k x\), where \(k\) is a positive constant, intersects \(H\) at the point \(P\)

1. Use calculus to determine, in terms of \(a\) and \(k\), an equation for the tangent to \(H\) at \(P\) The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\)
2. Determine the coordinates of \(A\) and the coordinates of \(B\), giving your answers in terms of \(a\) and \(k\)
3. Hence show that the area of triangle \(A O B\), where \(O\) is the origin, is independent of \(k\)

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 $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$

1. The point \(\mathrm { P } \left( 6 \mathrm { t } , \frac { 6 } { \mathrm { t } } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\) (a) 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 \(( - 9,12 )\).
(b) Find the coordinates of \(A\) and \(B\).

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 \(I\) meets the directrix of the parabola.

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.

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 \(B\).

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 \(B\).

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 \(P S N\).

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.

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where c is a positive constant. The point \(A\) on \(H\) has \(x\)-coordinate \(3 c\).

1. Write down the \(y\)-coordinate of \(A\).
2. Show that an equation of the normal to \(H\) at \(A\) is $$3 y = 27 x - 80 c$$ The normal to \(H\) at \(A\) meets \(H\) again at the point \(B\).
3. Find, in terms of \(c\), the coordinates of \(B\).

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 \(P Q R\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ Given that \(C\) meets the initial line at the point \(A\), as shown in Figure 1,

1. write down the polar coordinates of \(A\). The line \(l _ { 1 }\), also shown in Figure 1, is the tangent to \(C\) at the point \(B\) and is parallel to the initial line.
2. Use calculus to determine the polar coordinates of \(B\). The line \(l _ { 2 }\), also shown in Figure 1, is the tangent to \(C\) at \(A\) and is perpendicular to the initial line. The region \(R\), shown shaded in Figure 1, is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\)
3. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(p \sqrt { 3 } + q \pi\) where \(p\) and \(q\) are constants to be determined.

5. A curve has equation $$y ^ { 2 } = y \mathrm { e } ^ { - 2 x } - 3 x$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { - 2 x } + 3 } { \mathrm { e } ^ { - 2 x } - 2 y }$$ The curve crosses the \(y\)-axis at the origin and at the point \(P\).
   The tangent to the curve at the origin and the tangent to the curve at \(P\) meet at the point \(R\).
2. Find the coordinates of \(R\). \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-17_2644_1838_121_116}

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$16 x ^ { 3 } - 9 k x ^ { 2 } y + 8 y ^ { 3 } = 875$$ where \(k\) is a constant.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 k x y - 16 x ^ { 2 } } { 8 y ^ { 2 } - 3 k x ^ { 2 } }$$ Given that the curve has a turning point at \(x = \frac { 5 } { 2 }\)
2. find the value of \(k\)

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has parametric equations $$x = \sin t - 3 \cos ^ { 2 } t \quad y = 3 \sin t + 2 \cos t \quad 0 \leqslant t \leqslant 5$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) where \(t = \pi\) The point \(P\) lies on \(C\) where \(t = \pi\)
2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. Given that the tangent to the curve at \(P\) cuts \(C\) at the point \(Q\)
3. show that the value of \(t\) at point \(Q\) satisfies the equation $$9 \cos ^ { 2 } t + 2 \cos t - 7 = 0$$
4. Hence find the exact value of the \(y\) coordinate of \(Q\)

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$2 ^ { x } - 4 x y + y ^ { 2 } = 13 \quad y \geqslant 0$$ The point \(P\) lies on \(C\) and has \(x\) coordinate 2

1. Find the \(y\) coordinate of \(P\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The tangent to \(C\) at \(P\) crosses the \(x\)-axis at the point \(Q\).
3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(\frac { a \ln 2 + b } { c \ln 2 + d }\) where \(a , b , c\) and \(d\) are integers to be found.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = t + \frac { 1 } { t } \quad y = t - \frac { 1 } { t } \quad t > 0.7$$ The curve \(C\) intersects the \(x\)-axis at the point \(Q\).

1. Find the \(x\) coordinate of \(Q\). The line \(l\) is the normal to \(C\) at the point \(P\) as shown in Figure 2.
   Given that \(t = 2\) at \(P\)
2. write down the coordinates of \(P\)
3. Using calculus, show that an equation of \(l\) is $$3 x + 5 y = 15$$ The region, \(R\), shown shaded in Figure 2 is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
4. Using algebraic integration, find the exact volume of the solid of revolution formed when the region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with parametric equations $$x = 2 t ^ { 2 } - 6 t , \quad y = t ^ { 3 } - 4 t , \quad t \in \mathbb { R }$$ The curve cuts the \(x\)-axis at the origin and at the points \(A\) and \(B\), as shown in Figure 2.

1. Find the coordinates of \(A\) and show that \(B\) has coordinates (20, 0).
2. Show that the equation of the tangent to the curve at \(B\) is $$7 y + 4 x - 80 = 0$$ The tangent to the curve at \(B\) cuts the curve again at the point \(P\).
3. Find, using algebra, the \(x\) coordinate of \(P\).

1. The curve \(C\) has equation

$$2 x - 4 y ^ { 2 } + 3 x ^ { 2 } y = 4 x ^ { 2 } + 8$$ The point \(P ( 3,2 )\) lies on \(C\).
Find the equation of the normal to \(C\) at the point \(P\), writing your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers to be found.

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 5 + 2 \tan t \quad y = 8 \sec ^ { 2 } t \quad - \frac { \pi } { 3 } \leqslant t \leqslant \frac { \pi } { 4 }$$

1. Use parametric differentiation to find the gradient of \(C\) at \(x = 3\) The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where f is a quadratic function.
2. Find \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants to be found.
3. Find the range of f.