Find normal equation at point - Implicit equations and differentiation

Edexcel Paper 2 2023 June Q7

7 marks Standard +0.3

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.
A curve has equation $$x ^ { 3 } + 2 x y + 3 y ^ { 2 } = 47$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$ The point $P ( - 2,5 )$ lies on the curve.
Find the equation of the normal to the curve at $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.

Edexcel Paper 2 2024 June Q15

12 marks Challenging +1.2

The curve $C$ has equation

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

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The point $P ( 1,0 )$ lies on $C$.
Show that the normal to $C$ at $P$ has equation $$y = - 2 x + 2$$
Prove that the normal to $C$ at $P$ does not meet $C$ again. You should use algebra for your proof and make your reasoning clear.

Edexcel C4 Q1

8 marks Moderate -0.3

The curve $C$ has equation $5 x ^ { 2 } + 2 x y - 3 y ^ { 2 } + 3 = 0$. The point $P$ on the curve $C$ has coordinates $( 1,2 )$.
1. Find the gradient of the curve at $P$.
2. Find the equation of the normal to the curve $C$ at $P$, in the form $y = a x + b$, where $a$ and $b$ are constants.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6e307391-198f-4ea9-99ed-6ef184fca0f7-2_674_895_900_392}
\end{figure} In Fig. 1, the curve $C$ has equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = x + \frac { 2 } { x ^ { 2 } } , \quad x > 0$$ The shaded region is bounded by $C$, the $x$-axis and the lines with equations $x = 1$ and $x = 2$. The shaded region is rotated through $2 \pi$ radians about the $x$-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form $\pi ( a + \ln b )$, where $a$ and $b$ are constants.
(8)

Edexcel C4 Q3

8 marks Standard +0.3

3. 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 $( - 1 , - 3 )$. (8)
3. continued

Edexcel C4 Q3

11 marks Standard +0.3

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

Show that the normal to the curve at $P$ has the equation $y = 2 - x$.
Find the coordinates of the point where the normal to the curve at $P$ meets the curve again.
3. continued

Edexcel C4 Q2

8 marks Standard +0.3

2. 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.

WJEC Unit 3 Specimen Q11

11 marks Standard +0.3

11. (a) The curve $C$ is given by the equation $$x ^ { 4 } + x ^ { 2 } y + y ^ { 2 } = 13$$ Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point ( $- 1,3$ ).
(b) Show that the equation of the normal to the curve $y ^ { 2 } = 4 x$ at the point $P \left( p ^ { 2 } , 2 p \right)$ is $$y + p x = 2 p + p ^ { 3 }$$ Given that $p \neq 0$ and that the normal at $P$ cuts the $x$-axis at $B ( b , 0 )$, show that $b > 2$.

OCR MEI Paper 2 2023 June Q15

7 marks Standard +0.3

15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where $y = 1$, giving your answer in the form $\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0$, where $a , b$ and $c$ are constants to be found.

AQA Paper 1 2021 June Q12

8 marks Moderate -0.3

12 The equation of a curve is $$( x + y ) ^ { 2 } = 4 y + 2 x + 8$$ The curve intersects the positive $x$-axis at the point $P$.
12

Show that the gradient of the curve at $P$ is $- \frac { 3 } { 2 }$
12
Find the equation of 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.
[2 marks]
$\_\_\_\_$

12	Find the equation of 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.
	[2 marks]
	\(\_\_\_\_\)