Intersection existence or conditions

2 A line has equation \(y = 2 c x + 3\) and a curve has equation \(y = c x ^ { 2 } + 3 x - c\), where \(c\) is a constant.
Showing all necessary working, determine which of the following statements is correct.
A The line and curve intersect only for a particular set of values of \(c\).
B The line and curve intersect for all values of \(c\).
C The line and curve do not intersect for any values of \(c\).

9. (a) On separate axes sketch the graphs of

1. \(y = - 3 x + c\), where \(c\) is a positive constant,
2. \(y = \frac { 1 } { x } + 5\) On each sketch show the coordinates of any point at which the graph crosses the \(y\)-axis and the equation of any horizontal asymptote. Given that \(y = - 3 x + c\), where \(c\) is a positive constant, meets the curve \(y = \frac { 1 } { x } + 5\) at two distinct points,
   (b) show that \(( 5 - c ) ^ { 2 } > 12\) (c) Hence find the range of possible values for \(c\).

3 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).

9 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).

7

1. In this question you must show detailed reasoning. Find the range of values of the constant \(m\) for which the simultaneous equations \(y = m x\) and \(x ^ { 2 } + y ^ { 2 } - 6 x - 2 y + 5 = 0\) have real solutions.
2. Give a geometrical interpretation of the solution in the case where \(m = 2\).

11 In this question you must show detailed reasoning.
Determine for what values of \(k\) the graphs \(y = 2 x ^ { 2 } - k x\) and \(y = x ^ { 2 } - k\) intersect.

Show that the curve with equation \(x^2 - 3xy - 40 = 0\) and the line with equation \(3x + y + k = 0\) meet for all values of the constant \(k\). [5]