Tangent/normal with axis intercepts

10. The curve \(C\) satisfies the equation $$x \mathrm { e } ^ { 5 - 2 y } - y = 0 \quad x > 0 , \quad y > 0$$ The point \(P\) with coordinates ( \(2 \mathrm { e } ^ { - 1 } , 2\) ) lies on \(C\).
The tangent to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and cuts the \(y\)-axis at the point \(B\).
Given that \(O\) is the origin, find the exact area of triangle \(O A B\), giving your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}]{a377da06-a968-438c-bec2-ae55283dae47-35_4_21_127_2042} L

1. The curve \(C\) has equation

$$2 x ^ { 2 } y + 2 x + 4 y - \cos ( \pi y ) = 17$$

1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(\left( 3 , \frac { 1 } { 2 } \right)\) lies on \(C\).
   The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
2. Find the \(x\) coordinate of \(A\), giving your answer in the form \(\frac { a \pi + b } { c \pi + d }\), where \(a , b , c\) and \(d\) are integers to be determined.

4. The curve \(C\) has equation $$4 x ^ { 2 } - y ^ { 3 } - 4 x y + 2 ^ { y } = 0$$ The point \(P\) with coordinates \(( - 2,4 )\) lies on \(C\).

1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). The normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
2. Find the \(y\) coordinate of \(A\), giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are constants to be determined.
   (3)

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

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { - 2 x } + 3 } { \mathrm { e } ^ { - 2 x } - 2 y }$$ The curve crosses the \(y\)-axis at the origin and at the point \(P\).
   The tangent to the curve at the origin and the tangent to the curve at \(P\) meet at the point \(R\).
2. Find the coordinates of \(R\). \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-17_2644_1838_121_116}

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$2 ^ { x } - 4 x y + y ^ { 2 } = 13 \quad y \geqslant 0$$ The point \(P\) lies on \(C\) and has \(x\) coordinate 2

1. Find the \(y\) coordinate of \(P\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The tangent to \(C\) at \(P\) crosses the \(x\)-axis at the point \(Q\).
3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(\frac { a \ln 2 + b } { c \ln 2 + d }\) where \(a , b , c\) and \(d\) are integers to be found.

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$3 x ^ { 2 } + 2 y ^ { 2 } - 4 x y + 8 ^ { x } - 11 = 0$$ The point \(P\) has coordinates ( 1,2 ).

1. Verify that \(P\) lies on \(C\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at a point \(Q\).
3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers.

The curve \(C\) has the equation $$\sin(\pi y) - y - x^2 y = -5, \quad x > 0$$

1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5] The point \(P\) with coordinates \((2, 1)\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
2. Find the exact value of the \(x\)-coordinate of \(A\). [4]