Find vertical tangent points

7 The equation of a curve is $$y ^ { 3 } + 4 x y = 16$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 y } { 3 y ^ { 2 } + 4 x }\).
2. Show that the curve has no stationary points.
3. Find the coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.

7 The equation of a curve is \(\ln ( x y ) - y ^ { 3 } = 1\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x \left( 3 y ^ { 3 } - 1 \right) }\).
2. Find the coordinates of the point where the tangent to the curve is parallel to the \(y\)-axis, giving each coordinate correct to 3 significant figures.

5 The equation of a curve is \(x ^ { 2 } y - a y ^ { 2 } = 4 a ^ { 3 }\), where \(a\) is a non-zero constant.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x y } { 2 a y - x ^ { 2 } }\).
2. Hence find the coordinates of the points where the tangent to the curve is parallel to the \(y\)-axis. [4]

7 The equation of a curve is \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 5\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + y ^ { 2 } } { y ^ { 2 } - 2 x y }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(y\)-axis.

9 The equation of a curve is \(y \mathrm { e } ^ { 2 x } - y ^ { 2 } \mathrm { e } ^ { x } = 2\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { x } - y ^ { 2 } } { 2 y - \mathrm { e } ^ { x } }\).
2. Find the exact coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$x = \frac { 2 y ^ { 2 } + 6 } { 3 y - 3 }$$

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) giving your answer as a fully simplified fraction. The tangents at points \(P\) and \(Q\) on the curve are parallel to the \(y\)-axis, as shown in Figure 4.
2. Use the answer to part (a) to find the equations of these two tangents.

7. A curve is described by the equation $$x ^ { 2 } + 4 x y + y ^ { 2 } + 27 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). A point \(Q\) lies on the curve.
   The tangent to the curve at \(Q\) is parallel to the \(y\)-axis.
   Given that the \(x\) coordinate of \(Q\) is negative,
2. use your answer to part (a) to find the coordinates of \(Q\).

6 Fig. 6 shows the curve \(\mathrm { e } ^ { 2 y } = x ^ { 2 } + y\). \begin{figure}[h] \end{figure}

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x } { 2 \mathrm { e } ^ { 2 y } - 1 }\).
2. Hence find to 3 significant figures the coordinates of the point P , shown in Fig. 6, where the curve has infinite gradient. Section B (36 marks)

5 Fig. 6 shows the curve \(\mathrm { e } ^ { 2 y } = x ^ { 2 } + y\). \begin{figure}[h] \end{figure}

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x } { 2 \mathrm { e } ^ { 2 y } - 1 }\).
2. Hence find to 3 significant figures the coordinates of the point P , shown in Fig. 6, where the curve has infinite gradient.

6 \includegraphics[max width=\textwidth, alt={}, center]{02e31b5d-10dd-42b1-885a-6db610d788c3-2_570_1191_1509_420} The diagram shows the curve with equation \(x ^ { 2 } + y ^ { 3 } - 8 x - 12 y = 4\). At each of the points \(P\) and \(Q\) the tangent to the curve is parallel to the \(y\)-axis. Find the coordinates of \(P\) and \(Q\).

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(x ^ { 2 } + y ^ { 3 } - 10 x - 12 y - 5 = 0\) a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 10 - 2 x } { 3 y ^ { 2 } - 12 }\) At each of the points \(P\) and \(Q\) the tangent to the curve is parallel to the \(y\)-axis.
b. Find the exact coordinates of \(Q\).

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(x ^ { 2 } - 2 x y + 3 y ^ { 2 } = 50\)

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x } { 3 y - x }\) The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km .
   The points \(P\) and \(Q\) represent points that are furthest west and furthest east of the origin \(O\), as shown in Figure 4. Using part (a),
2. find the exact coordinates of the point \(P\).
3. Explain briefly how to find the coordinates of the point that is furthest north of the origin \(O\). (You do not need to carry out this calculation).

6 In this question you must show detailed reasoning. The diagram shows the curve with equation \(4 x y = 2 \left( x ^ { 2 } + 4 y ^ { 2 } \right) - 9 x\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 4 y - 9 } { 4 x - 16 y }\). At the point \(P\) on the curve the tangent to the curve is parallel to the \(y\)-axis and at the point \(Q\) on the curve the tangent to the curve is parallel to the \(x\)-axis.
2. Show that the distance \(P Q\) is \(k \sqrt { 5 }\), where \(k\) is a rational number to be determined.