1.07m Tangents and normals: gradient and equations

1. A curve has the equation

$$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point \(( 2 , - 4 )\) has the equation \(y = 3 x - 10\). (8)

14
2 \end{array} \right) , $$ and\\ where \(a\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.\\ Given that the two lines intersect,\\

1. find the position vector of their point of intersection,
2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.\\ 4. continued\\ 5. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
2. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
3. Find the coordinates of \(Q\).\\ 5. continued\\ 6. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).

\\

1. Write down a differential equation connecting \(N\) and \(t\). Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
2. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
3. find the value of \(k\),
4. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.\\ 6. continued\\ 7. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$

1. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
2. find a cartesian equation for the curve.
3. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
   7. continued
   7. continued

7 The function f is defined for all real numbers by $$f ( x ) = \sin \left( x + \frac { \pi } { 6 } \right)$$

1. Find the general solution of the equation \(\mathrm { f } ( x ) = 0\).
2. The quadratic function g is defined for all real numbers by $$\mathrm { g } ( x ) = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 4 } x ^ { 2 }$$ It can be shown that \(\mathrm { g } ( x )\) gives a good approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
   1. Show that \(\mathrm { g } ( 0.05 )\) and \(\mathrm { f } ( 0.05 )\) are identical when rounded to four decimal places.
   2. A chord joins the points on the curve \(y = \mathrm { g } ( x )\) for which \(x = 0\) and \(x = h\). Find an expression in terms of \(h\) for the gradient of this chord.
   3. Using your answer to part (b)(ii), find the value of \(\mathrm { g } ^ { \prime } ( 0 )\).

6 The diagram shows a circle \(C\) and a line \(L\), which is the tangent to \(C\) at the point \(( 1,1 )\). The equations of \(C\) and \(L\) are $$x ^ { 2 } + y ^ { 2 } = 2 \text { and } x + y = 2$$ respectively. \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_760_1301_552_395} The circle \(C\) is now transformed by a stretch with scale factor 2 parallel to the \(x\)-axis. The image of \(C\) under this stretch is an ellipse \(E\).

1. On the diagram below, sketch the ellipse \(E\), indicating the coordinates of the points where it intersects the coordinate axes.
2. Find equations of:
   1. the ellipse \(E\);
   2. the tangent to \(E\) at the point \(( 2,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_743_1301_1921_420}

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\). \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.

• \(a = - 2\)
• \(a = - 1\)
• \(a = 0\)
• State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
• State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
• 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
  2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).

Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).

With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which

Find its coordinates and fully justify that it is a cusp.

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

The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$p ^ { 3 } x - p y + c \left( 1 - p ^ { 4 } \right) = 0$$ The normal to \(H\) at the point \(P\) meets \(H\) again at the point \(Q\).
2. Find the coordinates of the midpoint of \(P Q\) in terms of \(c\) and \(p\), simplifying your answer where possible.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the rectangular hyperbola \(H\) with equation $$x y = c ^ { 2 } \quad x > 0$$ where \(c\) is a positive constant.
The point \(P \left( c t , \frac { c } { t } \right)\) lies on \(H\).
The line \(l\) is the tangent to \(H\) at the point \(P\).
The line \(l\) crosses the \(x\)-axis at the point \(A\) and crosses the \(y\)-axis at the point \(B\).
The region \(R\), shown shaded in Figure 2, is bounded by the \(x\)-axis, the \(y\)-axis and the line \(l\). Given that the length \(O B\) is twice the length of \(O A\), where \(O\) is the origin, and that the area of \(R\) is 32 , find the exact coordinates of the point \(P\).

1. The point \(P \left( a p ^ { 2 } , 2 a p \right)\), where \(a\) is a positive constant, lies on the parabola with equation

$$y ^ { 2 } = 4 a x$$ The normal to the parabola at \(P\) meets the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\)

1. Show that $$q = \frac { - p ^ { 2 } - 2 } { p }$$
2. Hence show that $$P Q ^ { 2 } = \frac { k a ^ { 2 } } { p ^ { 4 } } \left( p ^ { 2 } + 1 \right) ^ { n }$$ where \(k\) and \(n\) are integers to be determined.

1. The parabola \(C\) has equation \(y ^ { 2 } = 10 x\)

The point \(F\) is the focus of \(C\).

1. Write down the coordinates of \(F\). The point \(P\) on \(C\) has \(y\) coordinate \(q\), where \(q > 0\)
2. Show that an equation for the tangent to \(C\) at \(P\) is given by $$10 x - 2 q y + q ^ { 2 } = 0$$ The tangent to \(C\) at \(P\) intersects the directrix of \(C\) at the point \(A\).
   The point \(B\) lies on the directrix such that \(P B\) is parallel to the \(x\)-axis.
3. Show that the point of intersection of the diagonals of quadrilateral \(P B A F\) always lies on the \(y\)-axis.

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

The point \(P \left( a t ^ { 2 } , 2 a t \right) , t \neq 0\), lies on \(C\) The normal to \(C\) at \(P\) is parallel to the line with equation \(y = 2 x\)

1. For the point \(P\), show that \(t = - 2\) The normal to \(C\) at \(P\) intersects \(C\) again when \(x = 9\)
2. Determine the value of \(a\), giving a reason for your answer.

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

The point \(P\) on \(C\) has \(y\) coordinate \(p\), where \(p\) is a positive constant.

1. Show that an equation of the tangent to \(C\) at \(P\) is given by $$2 p y = 16 x + p ^ { 2 }$$ $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
2. Write down the equation of the directrix of \(C\). The line \(l\) is the reflection of the tangent to \(C\) at \(P\) in the directrix of \(C\).
   Given that \(l\) passes through the focus of \(C\),
3. determine the exact value of \(p\).

5. \begin{figure}[h] \end{figure} Diagram not drawn to scale $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$ The parabola C has equation \(\mathrm { y } ^ { 2 } = 16 \mathrm { x }\)

1. Deduce that the point \(\mathrm { P } \left( 4 \mathrm { p } ^ { 2 } , 8 \mathrm { p } \right)\) is a general point on C . The line I is the tangent to C at the point P .
2. Show that an equation for I is $$p y = x + 4 p ^ { 2 }$$ The finite region R , shown shaded in Figure 2, is bounded by the line I , the x -axis and the parabola C.
   The line \(I\) intersects the directrix of \(C\) at the point \(B\), where the \(y\) coordinate of \(B\) is \(\frac { 10 } { 3 }\) Given that \(\mathrm { p } > 0\)
3. show that the area of R is 36 \section*{Q uestion 5 continued}

1. The curve \(C\) has equation

$$y = \arccos \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2$$

1. Show that \(C\) has no stationary points. The normal to \(C\), at the point where \(x = 1\), crosses the \(x\)-axis at the point \(A\) and crosses the \(y\)-axis at the point \(B\). Given that \(O\) is the origin,
2. show that the area of the triangle \(O A B\) is \(\frac { 1 } { 54 } \left( p \sqrt { 3 } + q \pi + r \sqrt { 3 } \pi ^ { 2 } \right)\) where \(p\), \(q\) and \(r\) are integers to be determined.
   (5)

1. The parabola \(C\) has equation

$$y ^ { 2 } = 16 x$$ The distinct points \(P \left( p ^ { 2 } , 4 p \right)\) and \(Q \left( q ^ { 2 } , 4 q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0\) The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) meet at the point \(R ( - 28,6 )\).
Show that the area of triangle \(P Q R\) is 1331

1. The points \(P \left( 9 p ^ { 2 } , 18 p \right)\) and \(Q \left( 9 q ^ { 2 } , 18 q \right) , p \neq q\), lie on the parabola \(C\) with equation

$$y ^ { 2 } = 36 x$$ The line \(l\) passes through the points \(P\) and \(Q\)

1. Show that an equation for the line \(l\) is $$( p + q ) y = 2 ( x + 9 p q )$$ The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(A\).
2. Show that the coordinates of \(A\) are $$\left( 9 \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - 9 p q ( p + q ) \right)$$ Given that the points \(P\) and \(Q\) vary such that \(l\) always passes through the point \(( 12,0 )\)
3. find, in the form \(y ^ { 2 } = \mathrm { f } ( x )\), an equation for the locus of \(A\), giving \(\mathrm { f } ( x )\) in simplest form.

1. The parabola \(C\) has equation

$$y ^ { 2 } = 32 x$$ and the hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 9 } = 1$$

1. Write down the equations of the asymptotes of \(H\). The line \(l _ { 1 }\) is normal to \(C\) and parallel to the asymptote of \(H\) with positive gradient. The line \(l _ { 2 }\) is normal to \(C\) and parallel to the asymptote of \(H\) with negative gradient.
2. Determine
   1. an equation for \(l _ { 1 }\)
   2. an equation for \(l _ { 2 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet \(H\) at the points \(P\) and \(Q\) respectively.
3. Find the area of the triangle \(O P Q\), where \(O\) is the origin.

The rectangular hyperbola \(H\) has equation \(x y = 36\)

1. Use calculus to show that the equation of the tangent to \(H\) at the point \(P \left( 6 t , \frac { 6 } { t } \right)\) is $$y t ^ { 2 } + x = 12 t$$ The point \(Q \left( 12 t , \frac { 3 } { t } \right)\) also lies on \(H\).
2. Find the equation of the tangent to \(H\) at the point \(Q\). The tangent at \(P\) and the tangent at \(Q\) meet at the point \(R\).
3. Show that as \(t\) varies the locus of \(R\) is also a rectangular hyperbola.

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

The point \(A \left( a t ^ { 2 } , 2 a t \right)\), where \(t \neq 0\), lies on \(P\).

1. Use calculus to show that an equation of the tangent to \(P\) at \(A\) is $$y t = x + a t ^ { 2 }$$ The point \(B \left( 2 k ^ { 2 } , 4 k \right)\) and the point \(C \left( 2 k ^ { 2 } , - 4 k \right)\), where \(k\) is a constant, lie on \(P\).
   The tangent to \(P\) at \(B\) and the tangent to \(P\) at \(C\) intersect at the point \(D\).
   Given that the area of the triangle \(B C D\) is 432
2. determine the coordinates of \(B\) and the coordinates of \(C\).

1. The normal to the parabola \(y ^ { 2 } = 4 a x\) at the point \(P \left( a p ^ { 2 } , 2 a p \right)\) passes through the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\).

The line \(O P\) is perpendicular to the line \(O Q\), where \(O\) is the origin.
Prove that \(p ^ { 2 } = 2\)

1. \(P\) and \(Q\) are two distinct points on the ellipse described by the equation \(x ^ { 2 } + 4 y ^ { 2 } = 4\)

The line \(l\) passes through the point \(P\) and the point \(Q\).
The tangent to the ellipse at \(P\) and the tangent to the ellipse at \(Q\) intersect at the point \(( r , s )\).
Show that an equation of the line \(l\) is $$4 s y + r x = 4$$