1.07m Tangents and normals: gradient and equations

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.

5. The rectangular hyperbola \(H\) has equation \(x y = 9\) The point \(A\) on \(H\) has coordinates \(\left( 6 , \frac { 3 } { 2 } \right)\).

1. Show that the normal to \(H\) at the point \(A\) has equation $$2 y - 8 x + 45 = 0$$ The normal at \(A\) meets \(H\) again at the point \(B\).
2. Find the coordinates of \(B\).

9. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\)

1. Show that an equation of the normal to \(H\) at the point \(P \left( 5 p , \frac { 5 } { p } \right) , p \neq 0\), is $$y - p ^ { 2 } x = \frac { 5 } { p } - 5 p ^ { 3 }$$ This normal meets the line with equation \(y = - x\) at the point \(A\).
2. Show that the coordinates of \(A\) are $$\left( - \frac { 5 } { p } + 5 p , \frac { 5 } { p } - 5 p \right)$$ The point \(M\) is the midpoint of the line segment \(A P\).
   Given that \(M\) lies on the positive \(x\)-axis,
3. find the exact value of the \(x\) coordinate of point \(M\).
   

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 \(B\).

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 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 2 shows a sketch of the curve \(C\) with equation $$y ^ { 3 } - x ^ { 2 } + 4 x ^ { 2 } y = k$$ where \(k\) is a positive constant greater than 1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) lies on \(C\).
   Given that the normal to \(C\) at \(P\) has equation \(y = x\), as shown in Figure 2,
2. find the value of \(k\).

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 6 t - 3 \sin 2 t \quad y = 2 \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The curve meets the \(y\)-axis at 2 and the \(x\)-axis at \(k\), where \(k\) is a constant.

1. State the value of \(k\).
2. Use parametric differentiation to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t$$ where \(\lambda\) is a constant to be found. The point \(P\) with parameter \(\mathrm { t } = \frac { \pi } { 4 }\) lies on \(C\).
   The tangent to \(C\) at the point \(P\) cuts the \(y\)-axis at the point \(N\).
3. Find the exact \(y\) coordinate of \(N\), giving your answer in simplest form. The region bounded by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
4. 1. Show that the volume of this solid is given by $$\int _ { 0 } ^ { \alpha } \beta ( 1 - \cos 4 t ) d t$$ where \(\alpha\) and \(\beta\) are constants to be found.
   2. Hence, using algebraic integration, find the exact volume of this solid.

2. A curve \(C\) has the equation $$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation of the tangent to \(C\) at the point \(( 3 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y ^ { 2 } = 8 x\) and part of the line \(l\) with equation \(x = 18\) The region \(R\), shown shaded in Figure 2, is bounded by \(C\) and \(l\)

1. Show that the perimeter of \(R\) is given by $$\alpha + 2 \int _ { 0 } ^ { \beta } \sqrt { 1 + \frac { y ^ { 2 } } { 16 } } d y$$ where \(\alpha\) and \(\beta\) are positive constants to be determined.
2. Use the substitution \(y = 4 \sinh u\) and algebraic integration to determine the exact perimeter of \(R\), giving your answer in simplest form.

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),

1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).
   

2. An ellipse has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 2 \sin \theta ) , 0 < \theta < \frac { \pi } { 2 }\) The line \(L\) is a normal to the ellipse at the point \(P\).

1. Show that an equation for \(L\) is $$5 x \sin \theta - 2 y \cos \theta = 21 \sin \theta \cos \theta$$ Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(P Q\),
2. find the exact area of triangle \(O P M\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2 \theta\) WIHN SIHI NITIIUM ION OC
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The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) and the ellipse \(E\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) where \(a > b > 0\) The line \(l\) is a tangent to hyperbola \(H\) at the point \(P ( a \sec \theta , b \tan \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$b x \sec \theta - a y \tan \theta = a b$$ Given that the point \(F\) is the focus of ellipse \(E\) for which \(x > 0\) and that the line \(l\) passes through \(F\),
2. show that \(l\) is parallel to the line \(y = x\)

4. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l\) is a normal to \(H\) at the point \(P ( a \sec \theta , b \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\), and the point \(M\) is the midpoint of \(P Q\).
2. Find the coordinates of \(M\).
3. Hence find the cartesian equation of the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

8. The point \(\mathrm { P } ( 5 \sec \mathrm { u } , 3 \tan \mathrm { u } )\) lies on the hyperbola H with equation \(\frac { \mathrm { x } ^ { 2 } } { 25 } - \frac { \mathrm { y } ^ { 2 } } { 9 } = 1\). The tangent to \(H\) at \(P\) intersects the asymptote of \(H\) with equation \(y = \frac { 3 } { 5 } x\) at the point \(R\) and the asymptote with equation \(\mathrm { y } = - \frac { 3 } { 5 } \mathrm { x }\) at the point S .

1. Use differentiation to show that an equation of the tangent to H at P is $$3 x = 5 y \sin u + 15 \cos u$$
2. Prove that P is the mid-point of RS.

The curve \(C\) has equation \(y = k x ^ { 3 } - x ^ { 2 } + x - 5\), where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The point \(A\) with \(x\)-coordinate \(- \frac { 1 } { 2 }\) lies on \(C\). The tangent to \(C\) at \(A\) is parallel to the line with equation \(2 y - 7 x + 1 = 0\). Find
2. the value of \(k\),
3. the value of the \(y\)-coordinate of \(A\).

A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 4,9 )\). Given that $$f ^ { \prime } ( x ) = \frac { 3 \sqrt { } x } { 2 } - \frac { 9 } { 4 \sqrt { } x } + 2 , \quad x > 0$$

1. find \(\mathrm { f } ( x )\), giving each term in its simplest form. Point \(P\) lies on the curve. The normal to the curve at \(P\) is parallel to the line \(2 y + x = 0\)
2. Find the \(x\) coordinate of \(P\).