Equation of tangent or normal

11 The equation of a curve is \(y = 2 \sqrt { 3 x + 4 } - x\).

1. Find the equation of the normal to the curve at the point (4,4), giving your answer in the form \(y = m x + c\).
2. Find the coordinates of the stationary point.
3. Determine the nature of the stationary point.
4. Find the exact area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 The equation of a curve is $$y = k \sqrt { 4 x + 1 } - x + 5$$ where \(k\) is a positive constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the \(x\)-coordinate of the stationary point in terms of \(k\).
3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan ^ { - 1 } ( 2 )\) with the positive \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-08_534_462_248_804} The diagram shows part of the curve with equation \(y = \frac { 12 } { \sqrt [ 3 ] { 2 x + 1 } }\). The point \(A\) on the curve has coordinates \(\left( \frac { 7 } { 2 } , 6 \right)\).

1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = m x + c\). [4]
   \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-08_2716_38_109_2012}
2. Find the area of the region bounded by the curve and the lines \(x = 0 , x = \frac { 7 } { 2 }\) and \(y = 0\).

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\includegraphics[max width=\textwidth, alt={}, center]{d0074ac8-42d2-49f4-a417-4a348537bccc-4_521_809_258_669} The diagram shows part of the curve \(y = ( x - 2 ) ^ { 4 }\) and the point \(A ( 1,1 )\) on the curve. The tangent at \(A\) cuts the \(x\)-axis at \(B\) and the normal at \(A\) cuts the \(y\)-axis at \(C\).

1. Find the coordinates of \(B\) and \(C\).
2. Find the distance \(A C\), giving your answer in the form \(\frac { \sqrt { } a } { b }\), where \(a\) and \(b\) are integers.
3. Find the area of the shaded region.

4 A curve has equation \(y = \frac { 4 } { ( 3 x + 1 ) ^ { 2 } }\). Find the equation of the tangent to the curve at the point where the line \(x = - 1\) intersects the curve.

6 The line \(3 y + x = 25\) is a normal to the curve \(y = x ^ { 2 } - 5 x + k\). Find the value of the constant \(k\).

9 The point \(A ( 2,2 )\) lies on the curve \(y = x ^ { 2 } - 2 x + 2\).

1. Find the equation of the tangent to the curve at \(A\).
   The normal to the curve at \(A\) intersects the curve again at \(B\).
2. Find the coordinates of \(B\).
   The tangents at \(A\) and \(B\) intersect each other at \(C\).
3. Find the coordinates of \(C\).

1 A curve has equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 3 x - 4 x ^ { \frac { 1 } { 2 } } + 4\). Find the equation of the tangent to the curve at the point \(( 4,0 )\).

1. A curve has equation

$$y = 2 x ^ { 3 } - 5 x ^ { 2 } - \frac { 3 } { 2 x } + 7 \quad x > 0$$

1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\) lies on the curve and has \(x\) coordinate \(\frac { 1 } { 2 }\)
2. Find an equation of the normal to the curve at \(P\), writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

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1. The curve \(C\) has equation

$$y = \frac { x ^ { 2 } } { 3 } + \frac { 4 } { \sqrt { x } } + \frac { 8 } { 3 x } - 5 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(P ( 4,3 )\) lies on \(C\).
2. Find the equation of the normal to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

10. The curve \(C\) has equation $$x = 3 \sec ^ { 2 } 2 y \quad x > 3 \quad 0 < y < \frac { \pi } { 4 }$$

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Hence show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { q x \sqrt { x - 3 } }$$ where \(p\) is irrational and \(q\) is an integer, stating the values of \(p\) and \(q\).
3. Find the equation of the normal to \(C\) at the point where \(y = \frac { \pi } { 12 }\), giving your answer in the form \(y = m x + c\), giving \(m\) and \(c\) as exact irrational numbers.
   

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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 equation $$y = \frac { 16 } { 9 ( 3 x - k ) } \quad x \neq \frac { k } { 3 }$$ where \(k\) is a positive constant not equal to 3

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving your answer in simplest form in terms of \(k\). The point \(P\) with \(x\) coordinate 1 lies on \(C\).
   Given that the gradient of the curve at \(P\) is - 12
2. find the two possible values of \(k\) Given also that \(k < 3\)
3. find the equation of the normal to \(C\) at \(P\), writing your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
4. show, using algebraic integration that, $$\int _ { 1 } ^ { 3 } \frac { 16 } { 9 ( 3 x - k ) } d x = \lambda \ln 10$$ where \(\lambda\) is a constant to be found.

1. The curve \(C\) has equation

$$y = ( 3 x - 2 ) ^ { 6 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) Given that the point \(P \left( \frac { 1 } { 3 } , 1 \right)\) lies on \(C\),
2. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.

2. A curve \(C\) has equation $$y = \frac { 3 } { ( 5 - 3 x ) ^ { 2 } } , \quad x \neq \frac { 5 } { 3 }$$ The point \(P\) on \(C\) has \(x\)-coordinate 2. Find an equation of the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(x = \left( 9 + 16 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).
The curve crosses the \(x\)-axis at the point \(A\).

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

9. A curve has the equation \(y = x + \frac { 3 } { x } , x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate 1 .

1. Show that the gradient of the curve at \(P\) is - 2 .
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = m x + c\).
3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again.

1 Find the equation of the tangent to the curve \(y = \sqrt { 4 x + 1 }\) at the point ( 2,3 ).

3 Fig. 8 shows the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\). P is the point on this curve with \(x\)-coordinate 1 , and R is the point \(\left( 0 , - \frac { 7 } { 8 } \right)\). \begin{figure}[h] \end{figure}

1. Find the gradient of PR.
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that PR is a tangent to the curve.
3. Find the exact coordinates of the turning point Q .
4. Differentiate \(x \ln x - x\). Hence, or otherwise, show that the area of the region enclosed by the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is \(\frac { 59 } { 24 } - \frac { 1 } { 4 } \ln 2\).

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

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\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-3_590_606_1197_772} The diagram shows the curve with equation \(x = \left( 37 + 10 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).

1. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Hence find the equation of the tangent to the curve at the point ( 7,3 ), giving your answer in the form \(y = m x + c\).
3. Express \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
4. Hence
   (a) solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(8 \sin \theta - 6 \cos \theta = 9\),
   (b) find the greatest possible value of $$32 \sin x - 24 \cos x - ( 16 \sin y - 12 \cos y )$$ as the angles \(x\) and \(y\) vary.

12 A curve has equation \(y = 6 x \sqrt { x } + \frac { 32 } { x }\) for \(x > 0\)
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1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   [0pt] [4 marks]
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2. The point \(A\) lies on the curve and has \(x\)-coordinate 4
   Find the coordinates of the point where the tangent to the curve at \(A\) crosses the \(x\)-axis.
   [0pt] [5 marks]

10. \begin{figure}[h] \end{figure} In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 3 } x ^ { 2 } - 2 \sqrt { x } + 3 \quad x \geqslant 0$$ The point \(P\) lies on \(C\) and has \(x\) coordinate 4
The line \(l\) is the tangent to \(C\) at \(P\).

1. Show that \(l\) has equation $$13 x - 6 y - 26 = 0$$ The region \(R\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the curve \(C\), the line \(l\) and the \(x\)-axis.
2. Find the exact area of \(R\).

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\includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-06_823_588_260_242} The diagram shows the curve \(y = 1 - x + \frac { 6 } { \sqrt { x } }\) and the line \(l\), which is the normal to the curve at the point (1, 6).

1. Determine the equation of \(l\) in the form $$a x + b y = c$$ where \(a\), \(b\) and \(c\) are integers whose values are to be stated.
2. Verify that the curve intersects the \(x\)-axis at the point where \(x = 4\).
3. In this question you must show detailed reasoning. Determine the exact area of the shaded region enclosed between \(l\), the curve, the \(x\)-axis and the \(y\)-axis.

1. A curve has the equation \(y = \frac { x } { 2 } + 3 - \frac { 1 } { x } , x \neq 0\).

The point \(A\) on the curve has \(x\)-coordinate 2 .

1. Find the gradient of the curve at \(A\).
2. Show that the tangent to the curve at \(A\) has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
3. Find the coordinates of \(B\).