Line tangent to curve, find k for tangency

A question is this type if and only if a specific line y = mx + k (with m known) is to be a tangent to a given curve, and the task is to find the value of k using the discriminant condition, with no other unknown constant in the curve.

5 questions · Standard +0.0

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CAIE P1 2012 June Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_636_947_1738_598} The diagram shows the curve \(y = 7 \sqrt { } x\) and the line \(y = 6 x + k\), where \(k\) is a constant. The curve and the line intersect at the points \(A\) and \(B\).
  1. For the case where \(k = 2\), find the \(x\)-coordinates of \(A\) and \(B\).
  2. Find the value of \(k\) for which \(y = 6 x + k\) is a tangent to the curve \(y = 7 \sqrt { } x\).
Edexcel C1 2014 June Q9
9 marks Moderate -0.3
9. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 2 } + 8\) The line \(L\) has equation \(y = 3 x + k\), where \(k\) is a positive constant.
  1. Sketch \(C\) and \(L\) on separate diagrams, showing the coordinates of the points at which \(C\) and \(L\) cut the axes. Given that line \(L\) is a tangent to \(C\),
  2. find the value of \(k\).
OCR C1 2007 June Q10
12 marks Standard +0.3
10
  1. Solve the equation \(3 x ^ { 2 } - 14 x - 5 = 0\). A curve has equation \(\mathrm { y } = 3 \mathrm { x } ^ { 2 } - 14 \mathrm { x } - 5\).
  2. Sketch the curve, indicating the coordinates of all intercepts with the axes.
  3. Find the value of C for which the line \(\mathrm { y } = 4 \mathrm { x } + \mathrm { C }\) is a tangent to the curve.
Edexcel C1 2014 June Q11
10 marks Moderate -0.5
11. Given that $$f ( x ) = 2 x ^ { 2 } + 8 x + 3$$
  1. find the value of the discriminant of \(\mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(p ( x + q ) ^ { 2 } + r\) where \(p , q\) and \(r\) are integers to be found. The line \(y = 4 x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = \mathrm { f } ( x )\).
  3. Calculate the value of \(c\).
Edexcel C1 Q4
7 marks Standard +0.3
4. (a) Find in exact form the coordinates of the points where the curve \(y = x ^ { 2 } - 4 x + 2\) crosses the \(x\)-axis.
(b) Find the value of the constant \(k\) for which the straight line \(y = 2 x + k\) is a tangent to the curve \(y = x ^ { 2 } - 4 x + 2\).