Show specific gradient value

8 The equation of a curve is $$x ^ { 2 } + 2 x y - y ^ { 2 } + 8 = 0$$

1. Show that the tangent to the curve at the point \(( - 2,2 )\) is parallel to the \(x\)-axis.
2. Find the equation of the tangent to the curve at the other point on the curve for which \(x = - 2\), giving your answer in the form \(y = m x + c\).

7 The equation of a curve is \(x ^ { 2 } - 4 x y - 2 y ^ { 2 } = 1\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the gradient of the curve at the point \(( - 1,2 )\) is \(- \frac { 5 } { 2 }\). [5]
2. Show that the curve has no stationary points.
3. Find the \(x\)-coordinate of each of the points on the curve at which the tangent is parallel to the \(y\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2. The curve \(C\) has equation $$3 ^ { x - 1 } + x y - y ^ { 2 } + 5 = 0$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(( 1,3 )\) on the curve \(C\) can be written in the form \(\frac { 1 } { \lambda } \ln \left( \mu \mathrm { e } ^ { 3 } \right)\), where \(\lambda\) and \(\mu\) are integers to be found.

2. A curve has the equation $$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point \(( 2 , - 4 )\) has the equation \(y = 3 x - 10\).

1. A curve has the equation

$$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point \(( 2 , - 4 )\) has the equation \(y = 3 x - 10\). (8)