Tangent with given gradient

2 The line \(4 y = x + c\), where \(c\) is a constant, is a tangent to the curve \(y ^ { 2 } = x + 3\) at the point \(P\) on the curve.

1. Find the value of \(c\).
2. Find the coordinates of \(P\).

8 The equation of a curve is \(2 x ^ { 2 } - 3 x - 3 y + y ^ { 2 } = 6\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 3 } { 3 - 2 y }\).
2. Find the coordinates of the two points on the curve at which the gradient is - 1 .

7 The equation of a curve is \(3 x ^ { 2 } + 4 x y + 3 y ^ { 2 } = 5\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x + 2 y } { 2 x + 3 y }\).
2. Hence find the exact coordinates of the two points on the curve at which the tangent is parallel to \(y + 2 x = 0\).

1. A curve has equation

$$4 x ^ { 2 } - y ^ { 2 } + 2 x y + 5 = 0$$ The points \(P\) and \(Q\) lie on the curve.
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) at \(P\) and at \(Q\),

1. use implicit differentiation to show that \(y - 6 x = 0\) at \(P\) and at \(Q\).
2. Hence find the coordinates of \(P\) and \(Q\).

4. A curve has equation \(3 x ^ { 2 } - y ^ { 2 } + x y = 4\). The points \(P\) and \(Q\) lie on the curve. The gradient of the tangent to the curve is \(\frac { 8 } { 3 }\) at \(P\) and at \(Q\).

1. Use implicit differentiation to show that \(y - 2 x = 0\) at \(P\) and at \(Q\).
2. Find the coordinates of \(P\) and \(Q\).

8. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
2. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
3. Find the coordinates of \(Q\).

8 The points \(P\) and \(Q\) lie on the curve with equation $$2 x ^ { 2 } - 5 x y + y ^ { 2 } + 9 = 0$$ The tangents to the curve at \(P\) and \(Q\) are parallel, each having gradient \(\frac { 3 } { 8 }\).

1. Show that the \(x\) - and \(y\)-coordinates of \(P\) and \(Q\) are such that \(x = 2 y\).
2. Hence find the coordinates of \(P\) and \(Q\).

9 In this question you must show detailed reasoning.
The curve \(x y + y ^ { 2 } = 8\) is shown in Fig. 9. \begin{figure}[h] \end{figure} Find the coordinates of the points on the curve at which the normal has gradient 2.