Tangent with given gradient or condition

1 Find the value of the constant \(c\) for which the line \(y = 2 x + c\) is a tangent to the curve \(y ^ { 2 } = 4 x\).

9 The curve \(C _ { 1 }\) has equation \(y = x ^ { 2 } - 4 x + 7\). The curve \(C _ { 2 }\) has equation \(y ^ { 2 } = 4 x + k\), where \(k\) is a constant. The tangent to \(C _ { 1 }\) at the point where \(x = 3\) is also the tangent to \(C _ { 2 }\) at the point \(P\). Find the value of \(k\) and the coordinates of \(P\).

10 A straight line has equation \(y = - 2 x + k\), where \(k\) is a constant, and a curve has equation \(y = \frac { 2 } { x - 3 }\).

1. Show that the \(x\)-coordinates of any points of intersection of the line and curve are given by the equation \(2 x ^ { 2 } - ( 6 + k ) x + ( 2 + 3 k ) = 0\).
2. Find the two values of \(k\) for which the line is a tangent to the curve. The two tangents, given by the values of \(k\) found in part (ii), touch the curve at points \(A\) and \(B\).
3. Find the coordinates of \(A\) and \(B\) and the equation of the line \(A B\).