3. The figure below shows the curve with cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = x y\).
\includegraphics[max width=\textwidth, alt={}, center]{f42517a5-d7ed-40f3-bb04-faea97d4b19b-08_830_997_228_262}
- Show that the polar equation of the curve is \(r ^ { 2 } = a \sin b \theta\), where \(a\) and \(b\) are positive constants to be determined.
- Determine the exact maximum value of \(r\).
- Determine the area enclosed by one of the loops.
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\section*{4. In this question you must show detailed reasoning.}
- The curves with equations
$$y = \frac { 3 } { 4 } \sinh x \text { and } y = \tanh x + \frac { 1 } { 5 }$$
intersect at just one point \(P\)
- Use algebra to show that the \(x\) coordinate of \(P\) satisfies the equation
$$15 \mathrm { e } ^ { 4 x } - 48 \mathrm { e } ^ { 3 x } + 32 \mathrm { e } ^ { x } - 15 = 0$$
- Show that \(\mathrm { e } ^ { x } = 3\) is a solution of this equation.
- Hence state the exact coordinates of \(P\).
(ii) Show that
$$\int _ { - 4 } ^ { 0 } \frac { \mathrm { e } ^ { \frac { 1 } { x } } } { x ^ { 2 } } \mathrm {~d} x = \mathrm { e } ^ { - \frac { 1 } { 4 } }$$
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