Find normal line equation

5. A curve has equation $$y = \frac { x ^ { 3 } } { 6 } + 4 \sqrt { x } - 15 \quad x \geqslant 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in simplest form. The point \(P \left( 4 , \frac { 11 } { 3 } \right)\) lies on the curve.
2. Find the equation of the normal to the curve at \(P\). Write 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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13. The curve \(C\) has equation $$y = 3 x ^ { 2 } - 4 x + 2$$ The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)

1. Show that \(l _ { 1 }\) has equation $$x + 2 y - 3 = 0$$ The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
2. By solving simultaneous equations, determine the coordinates of the point \(Q\). Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
3. Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when $$k ^ { 2 } - 16 k + 40 = 0$$
4. Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where

$$f ^ { \prime } ( x ) = 3 \sqrt { x } - \frac { 9 } { \sqrt { x } } + 2$$ Given that the point \(P ( 9,14 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying your answer,
2. find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

9. The gradient of the curve \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 } .$$ The point \(P ( 1,4 )\) lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\).
2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
3. Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).

10. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
3. Show that the normal to the curve at \(P\) does not intersect the curve again.

6. The curve with equation \(y = x ^ { 2 } + 2 x\) passes through the origin, \(O\).

1. Find an equation for the normal to the curve at \(O\).
2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again.

9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).

1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).

9.
\includegraphics[max width=\textwidth, alt={}, center]{33f9663f-26bb-445e-af6e-ca5ca927f7dd-3_638_757_1064_493} The diagram shows the curve with equation \(y = 5 + x - x ^ { 2 }\) and the normal to the curve at the point \(P ( 1,5 )\).

1. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again.
3. Show that the area of the shaded region bounded by the curve and the straight line \(P Q\) is \(\frac { 4 } { 3 }\).

2 Find the equation of the normal to the curve \(y = 8 x ^ { 4 } + 4\) at the point where \(x = \frac { 1 } { 2 }\).

1. Find the equation of the tangent to the curve \(y = x ^ { 4 }\) at the point where \(x = 2\). Give your answer in the form \(y = m x + c\).
2. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 4 }\) where \(x = 2\) and \(x = 2.1\).
3. (A) Expand \(( 2 + h ) ^ { 4 }\).
   (B) Simplify \(\frac { ( 2 + h ) ^ { 4 } - 2 ^ { 4 } } { h }\).
   (C) Show how your result in part (iii) \(( B )\) can be used to find the gradient of \(y = x ^ { 4 }\) at the point where \(x = 2\).

5. A curve has the equation \(y = \sqrt { 3 x + 11 }\). The point \(P\) on the curve has \(x\)-coordinate 3 .

1. Show that the tangent to the curve at \(P\) has the equation $$3 x - 4 \sqrt { 5 } y + 31 = 0$$ The normal to the curve at \(P\) crosses the \(y\)-axis at \(Q\).
2. Find the \(y\)-coordinate of \(Q\) in the form \(k \sqrt { 5 }\).

10. The curve \(C\) has equation \(y = x ^ { 3 } - 5 x + \frac { 2 } { x } , x \neq 0\). The points \(A\) and \(B\) both lie on \(C\) and have coordinates \(( 1 , - 2 )\) and \(( - 1,2 )\) respectively.

1. Show that the gradient of \(C\) at \(A\) is equal to the gradient of \(C\) at \(B\).
2. Show that an equation for the normal to \(C\) at \(A\) is \(4 y = x - 9\). The normal to \(C\) at \(A\) meets the \(y\)-axis at the point \(P\). The normal to \(C\) at \(B\) meets the \(y\)-axis at the point \(Q\).
3. Find the length of \(P Q\).

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.

6 Find the equation of the normal to the curve \(y = \frac { 6 } { x ^ { 2 } } - 5\) at the point on the curve where \(x = 2\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

5 Find the equation of the normal to the curve \(y = 8 x ^ { 4 } + 4\) at the point where \(x = \frac { 1 } { 2 }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve \(C\) with equation \(y = 4 x ^ { 2 } - 10 x + 7\) and two straight lines, \(l _ { 1 }\) and \(l _ { 2 }\). The line \(l _ { 1 }\) is the normal to \(C\) at the point \(\left( \frac { 1 } { 2 } , 3 \right)\). The line \(l _ { 2 }\) is the normal to \(C\) at the minimum point of \(C\).

1. Determine the equation of \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined. The shaded region shown in the diagram is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\).
2. Determine the inequalities that define the shaded region, including its boundaries.

17 Show that, for the curve \(y = x ^ { 2 }\), the equation of the normal at the point \(\left( t , t ^ { 2 } \right)\) is \(y = - \frac { x } { 2 t } + t ^ { 2 } + \frac { 1 } { 2 }\), as given in line 27.

2 A curve has equation \(y = x ^ { 5 } - 6 x ^ { 3 } - 3 x + 25\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(P\) on the curve has coordinates \(( 2,3 )\).
   1. Show that the gradient of the curve at \(P\) is 5 .
   2. Hence find an equation of the normal to the curve at \(P\), expressing your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
3. Determine whether \(y\) is increasing or decreasing when \(x = 1\).

6 The curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{2f7a8e95-4994-4732-a9a4-306c7b6cad92-3_444_819_1434_609} The curve crosses the \(x\)-axis at the origin \(O\), and the point \(A ( 2 , - 6 )\) lies on the curve.

1. 1. Find the gradient of the curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) at the point \(A\).
   2. Hence find the equation of the normal to the curve at the point \(A\), giving your answer in the form \(x + p y + q = 0\), where \(p\) and \(q\) are integers.
2. 1. Find the value of \(\int _ { 0 } ^ { 2 } \left( 12 x ^ { 2 } - 19 x - 2 x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

7. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

3. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   [0pt] [P1 June 2003 Question 8*]

10. 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.

9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).

1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).

9. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
3. Show that the normal to the curve at \(P\) does not intersect the curve again.