Find normal equation at point

4 A curve has equation \(x ^ { 2 } y + 2 y ^ { 3 } = 48\).
Find the equation of the normal to the curve at the point ( 4,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

5 The equation of a curve is $$x ^ { 2 } - 2 x ^ { 2 } y + 3 y = 9$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 4 x y } { 2 x ^ { 2 } - 3 }\).
2. Find the equation of the normal to the curve at the point where \(x = 2\), giving your answer in the form \(a x + b y + c = 0\).

3 Find the equation of the normal to the curve $$x ^ { 2 } \ln y + 2 x + 5 y = 11$$ at the point \(( 3,1 )\).

3 A curve has equation $$3 \ln x + 6 x y + y ^ { 2 } = 16$$ Find the equation of the normal to the curve at the point \(( 1,2 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6 The equation of a curve is \(3 x ^ { 2 } + 4 x y + y ^ { 2 } = 24\). Find the equation of the normal to the curve at the point ( 1,3 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

4. The curve \(C\) has equation $$3 y \mathrm { e } ^ { - 2 x } = 4 x ^ { 2 } + y ^ { 2 } + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) on \(C\) has coordinates \(( 0,2 )\).
2. Find the equation of the normal to \(C\) at \(P\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
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2. A curve \(C\) has equation $$x ^ { 3 } - 4 x y + 2 x + 3 y ^ { 2 } - 3 = 0$$ Find an equation of the normal 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. \includegraphics[max width=\textwidth, alt={}, center]{c6bde466-61ec-437d-a3b4-84511a98d788-05_108_166_2612_1781}

3. The curve \(C\) has equation $$x = 2 \sin y .$$

1. Show that the point \(P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)\) lies on \(C\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }\) at \(P\).
3. Find an equation of the normal to \(C\) at \(P\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are exact constants.

4. The point \(P\) is the point on the curve \(x = 2 \tan \left( y + \frac { \pi } { 12 } \right)\) with \(y\)-coordinate \(\frac { \pi } { 4 }\). Find an equation of the normal to the curve at \(P\).

6. A curve has equation $$4 y ^ { 2 } + 3 x = 6 y \mathrm { e } ^ { - 2 x }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The curve crosses the \(y\)-axis at the origin and at the point \(P\).
2. Find the equation of the normal to the curve at \(P\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.

1. The curve \(C\) has the equation \(2 x + 3 y ^ { 2 } + 3 x ^ { 2 } y = 4 x ^ { 2 }\).

The point \(P\) on the curve has coordinates \(( - 1,1 )\).

1. Find the gradient of the curve at \(P\).
2. Hence find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. A curve \(C\) is described by the equation

$$3 x ^ { 2 } - 2 y ^ { 2 } + 2 x - 3 y + 5 = 0$$ Find an equation of the normal to \(C\) at the point ( 0,1 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

4. The curve \(C\) has the equation \(y \mathrm { e } ^ { - 2 x } = 2 x + y ^ { 2 }\).

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

5
- 1 \end{array} \right)$$ where \(a\) is a constant.\\ Given that \(\overrightarrow { O C }\) is perpendicular to \(\overrightarrow { B C }\)\\ (b) find the possible values of \(a\).

1. The curve \(C\) is defined by the equation

$$8 x ^ { 3 } - 3 y ^ { 2 } + 2 x y = 9$$ Find an equation of the normal to \(C\) at the point ( 2,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. 4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a segment \(P Q R P\) of a circle with centre \(O\) and radius 5 cm .
Given that \begin{itemize} \item angle \(P O R\) is \(\theta\) radians \item \(\theta\) is increasing, from 0 to \(\pi\), at a constant rate of 0.1 radians per second \item the area of the segment \(P Q R P\) is \(A \mathrm {~cm} ^ { 2 }\)

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 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } = 25\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { y }\).
2. Hence find the equation of the normal to the circle at the point ( 3,4 ).

4 Find the equation of the normal to the curve $$x ^ { 3 } + 4 x ^ { 2 } y + y ^ { 3 } = 6$$ at the point \(( 1,1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6 The equation of a curve is \(x ^ { 2 } + 3 x y + 4 y ^ { 2 } = 58\). Find the equation of the normal at the point \(( 2,3 )\) on the curve, giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. A curve has the equation

$$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point \(P\) on the curve has coordinates \(( - 1,3 )\).

1. Show that the normal to the curve at \(P\) has the equation \(y = 2 - x\).
2. Find the coordinates of the point where the normal to the curve at \(P\) meets the curve again.

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.

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 \(( - 1 , - 3 )\).

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 \(P\).

7 Find the equation of the normal to the curve \(x ^ { 3 } + 2 x ^ { 2 } y = y ^ { 3 } + 15\) at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5 A curve has equation \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 2 } + 1 = 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y - 3 x ^ { 2 } } { 2 y - 3 x ^ { 2 } }\).
2. Find the equation of the normal to the curve at the point ( 1,2 ).