1.07m Tangents and normals: gradient and equations

1. A curve has the equation

$$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation for the tangent to the curve at the point \(( 2 , - 2 )\).
   

6.

1. Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
2. Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
3. Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).

1. A curve has the equation

$$4 \cos x + 2 \sin y = 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y\).
2. Find an equation for the tangent to the curve at the point ( \(\frac { \pi } { 3 } , \frac { \pi } { 6 }\) ), giving your answer in the form \(a x + b y = c\), where \(a\) and \(b\) are integers.
   

14
2 \end{array} \right) , $$ where \(a\) is a constant and \(s\) and \(t\) 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.\\ 8. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$

1. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
2. Find the value which the population of the town will approach in the long term, according to the model.\\ 9. A curve has parametric equations $$x = t ( t - 1 ) , \quad y = \frac { 4 t } { 1 - t } , \quad t \neq 1$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) on the curve has parameter \(t = - 1\).
2. Show that the tangent to the curve at \(P\) has the equation $$x + 3 y + 4 = 0$$ The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
3. Find the coordinates of \(Q\).

2. 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. \includegraphics[max width=\textwidth, alt={}, center]{a86f277c-a2ec-4ba0-ab08-575cad2a5e53-3_424_698_246_479} The diagram shows the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\) where $$\mathrm { f } ( x ) = \frac { \cos x } { 2 - \sin x } , \quad x \in \mathbb { R }$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - 2 \sin x } { ( 2 - \sin x ) ^ { 2 } }\).
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = \pi\).
3. Find the minimum and maximum values of \(\mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\).
4. Explain why your answers to part (c) are the minimum and maximum values of \(\mathrm { f } ( x )\) for all real values of \(x\).

3. A curve has the equation $$3 x ^ { 2 } + x y - 2 y ^ { 2 } + 25 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( 1,4 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5.

1. Given that $$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } }$$
2. Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).

6. A curve has parametric equations $$x = \frac { t } { 2 - t } , \quad y = \frac { 1 } { 1 + t } , \quad - 1 < t < 2$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \left( \frac { 2 - t } { 1 + t } \right) ^ { 2 }\).
2. Find an equation for the normal to the curve at the point where \(t = 1\).
3. Show that the cartesian equation of the curve can be written in the form $$y = \frac { 1 + x } { 1 + 3 x }$$

3. A curve has the equation $$2 \sin 2 x - \tan y = 0$$

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 \(\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)\) has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 } .$$

4. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( - 1 , - 3 )\).

4

1. Show that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
2. Find \(\int _ { 2.5 } ^ { 3.9 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 9 } } \mathrm {~d} x\), giving your answer in the form \(a \ln b\), where \(a\) and \(b\) are rational numbers.
3. There are two points on the curve \(y = \frac { \cosh x } { 2 + \sinh x }\) at which the gradient is \(\frac { 1 } { 9 }\). Show that one of these points is \(\left( \ln ( 1 + \sqrt { 2 } ) , \frac { 1 } { 3 } \sqrt { 2 } \right)\), and find the coordinates of the other point, in a similar form.

4

1. Given that \(k \geqslant 1\) and \(\cosh x = k\), show that \(x = \pm \ln \left( k + \sqrt { k ^ { 2 } - 1 } \right)\).
2. Find \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x\), giving the answer in an exact logarithmic form.
3. Solve the equation \(6 \sinh x - \sinh 2 x = 0\), giving the answers in an exact form, using logarithms where appropriate.
4. Show that there is no point on the curve \(y = 6 \sinh x - \sinh 2 x\) at which the gradient is 5 .

4.A curve \(C\) has equation \(y = \mathrm { f } ( x )\) with \(\mathrm { f } ^ { \prime } ( x ) > 0\) .The \(x\)-coordinate of the point \(P\) on the curve is \(a\) .The tangent and the normal to \(C\) are drawn at \(P\) .The tangent cuts the \(x\)-axis at the point \(A\) and the normal cuts the \(x\)-axis at the point \(B\) .

1. Show that the area of \(\triangle A P B\) is $$\frac { 1 } { 2 } [ \mathrm { f } ( a ) ] ^ { 2 } \left( \frac { \left[ \mathrm { f } ^ { \prime } ( a ) \right] ^ { 2 } + 1 } { \mathrm { f } ^ { \prime } ( a ) } \right)$$
2. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 5 x }\) and the area of \(\triangle A P B\) is \(\mathrm { e } ^ { 5 a }\) ,find and simplify the exact value of \(a\) .

3.Given that \(\phi = \frac { 1 } { 2 } ( \sqrt { 5 } + 1 )\) ,

1. show that
   1. \(\phi ^ { 2 } = \phi + 1\)
   2. \(\frac { 1 } { \phi } = \phi - 1\)
2. The equations of two curves are $$\begin{array} { r l r l } y & = \frac { 1 } { x } & x > 0 \\ \text { and } & y & = \ln x - x + k & x > 0 \end{array}$$ where \(k\) is a positive constant.
   The curves touch at the point \(P\) .
   Find in terms of \(\phi\)
   1. the coordinates of \(P\) ,
   2. the value of \(k\) .

4.The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$

1. Show that \(m\) satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin \(O\) to \(C _ { 1 }\) touch \(C _ { 1 }\) at the points \(A\) and \(B\) .
2. Find the coordinates of the points \(A\) and \(B\) .
   (8)
   Another circle \(C _ { 2 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 13\) .The tangents from the point \(( 4 , - 7 )\) to \(C _ { 2 }\) touch it at the points \(P\) and \(Q\) .
3. Find the coordinates of either the point \(P\) or the point \(Q\) .
   (2)

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$ The lines \(x = 0\) and \(y = \frac { x } { 3 }\) are asymptotes to \(C _ { 1 }\). The point \(A\) on \(C _ { 1 }\) is a minimum and the point \(B\) on \(C _ { 1 }\) is a maximum.

1. Find the coordinates of \(A\) and \(B\). There is a normal to \(C _ { 1 }\), which does not intersect \(C _ { 1 }\) a second time, that has equation \(x = k\), where \(k > 0\).
2. Write down the value of \(k\). The point \(P ( \alpha , \beta ) , \alpha > 0\) and \(\alpha \neq k\), lies on \(C _ { 1 }\). The normal to \(C _ { 1 }\) at \(P\) does not intersect \(C _ { 1 }\) a second time.
3. Find the value of \(\alpha\), leaving your answer in simplified surd form.
4. Find the equation of this normal. The curve \(C _ { 2 }\) has equation \(y = | \mathrm { f } ( x ) |\)
5. Sketch \(C _ { 2 }\) stating the coordinates of any turning points and the equations of any asymptotes. The line with equation \(y = m x + 1\) does not touch or intersect \(C _ { 2 }\).
6. Find the set of possible values for \(m\).

10. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point (4,25). Given that $$f ^ { \prime } ( x ) = \frac { 3 } { 8 } x ^ { 2 } - 10 x ^ { - \frac { 1 } { 2 } } + 1 , \quad x > 0$$

1. find \(\mathrm { f } ( x )\), simplifying each term.
2. Find an equation of the normal to the curve at the point ( 4,25 ). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.

2. The point \(P\) with coordinates \(\left( \frac { \pi } { 2 } , 1 \right)\) lies on the curve with equation $$4 x \sin x = \pi y ^ { 2 } + 2 x , \quad \frac { \pi } { 6 } \leqslant x \leqslant \frac { 5 \pi } { 6 }$$ Find an equation of the normal to the curve at \(P\).

1. A rectangular hyperbola \(H\) has equation \(x y = 25\)

The point \(P \left( 5 t , \frac { 5 } { t } \right) , t \neq 0\), is a general point on \(H\).

1. Show that the equation of the tangent to \(H\) at \(P\) is \(t ^ { 2 } y + x = 10 t\) The distinct points \(Q\) and \(R\) lie on \(H\). The tangent to \(H\) at the point \(Q\) and the tangent to \(H\) at the point \(R\) meet at the point \(( 15 , - 5 )\).
2. Find the coordinates of the points \(Q\) and \(R\).

6. The parabola \(C\) has Cartesian equation \(y ^ { 2 } = 8 x\) The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) and the point \(Q \left( 2 q ^ { 2 } , 4 q \right)\), where \(p , q \neq 0 , p \neq q\), are points on \(C\).

1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = 2 p ^ { 3 } + 4 p$$
2. Write down an equation of the normal to \(C\) at \(Q\) The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(N\)
3. Show that \(N\) has coordinates $$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$ The line \(O N\), where \(O\) is the origin, is perpendicular to the line \(P Q\)
4. Find the value of \(( p + q ) ^ { 2 } - 3 p q\)

10 A curve has equation \(y = x ^ { 2 } + x\).

1. Find the gradient of the curve at the point for which \(x = 2\).
2. Find the equation of the normal to the curve at the point for which \(x = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find the values of \(k\) for which the line \(y = k x - 4\) is a tangent to the curve.

3 Find the equation of the normal to the curve \(y = x ^ { 3 } - 4 x ^ { 2 } + 7\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

8

1. Find the equation of the tangent to the curve \(y = 7 + 6 x - x ^ { 2 }\) at the point \(P\) where \(x = 5\), giving your answer in the form \(a x + b y + c = 0\).
2. This tangent meets the \(x\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).
3. Find the equation of the line of symmetry of the curve \(y = 7 + 6 x - x ^ { 2 }\).
4. State the set of values of \(x\) for which \(7 + 6 x - x ^ { 2 }\) is an increasing function.

10 A circle has centre \(C ( - 2,4 )\) and radius 5 .

1. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. Show that the tangent to the circle at the point \(P ( - 5,8 )\) has equation \(3 x - 4 y + 47 = 0\).
3. Verify that the point \(T ( 3,14 )\) lies on this tangent.
4. Find the area of the triangle \(C P T\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}