Find tangent equation at point - Implicit equations and differentiation

7 marks Standard +0.3

A curve has the equation $x = y \sqrt { 1 - 2 y }$.
1. Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point $A$ on the curve has $y$-coordinate - 1 .
Show that the equation of tangent to the curve at $A$ can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where $p$ and $q$ are integers to be found.

7 marks Standard +0.0

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

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

7 marks Standard +0.3

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

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y$.
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.

7 marks Standard +0.3

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

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in its simplest form in terms of $x$ and $y$.
Show that the tangent to the curve at the point $\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)$ has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 } .$$

8 marks Standard +0.3

3 The equation of a curve $C$ is $( x + 3 ) ( y + 4 ) = x ^ { 2 } + y ^ { 2 }$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
The line $2 y = x + 3$ meets $C$ at two points. What can be said about the tangents to $C$ at these points? Justify your answer.
Find the equation of the tangent at the point ( 6,0 ), giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.

8 marks Standard +0.3

4. A curve has the equation $x = y \sqrt { 1 - 2 y }$.

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point $A$ on the curve has $y$-coordinate - 1 .
Show that the equation of tangent to the curve at $A$ can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where $p$ and $q$ are integers to be found.

8 marks Moderate -0.3

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

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

8 marks Standard +0.3

5. A curve has the equation $$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$

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

8 marks Moderate -0.3

$$2 \sin 2 x - \tan y = 0$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in its simplest form in terms of $x$ and $y$.
Show that the tangent to the curve at the point $\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)$ has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 }$$
1. continued
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3cf64017-e982-4165-9885-8524aaabdf84-06_433_812_246_479} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with parametric equations $$x = a \sqrt { t } , \quad y = a t ( 1 - t ) , \quad t \geq 0$$ where $a$ is a positive constant.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. The curve meets the $x$-axis at the origin, $O$, and at the point $A$. The tangent to the curve at $A$ meets the $y$-axis at the point $B$ as shown.
Show that the area of triangle $O A B$ is $a ^ { 2 }$.