Line tangent to curve, find constant

A question is this type if and only if it states that a line is a tangent to a curve (both containing an unknown constant) and asks to find the value(s) of that constant by setting the discriminant of the combined equation equal to zero.

5 questions · Standard +0.1

1.07m Tangents and normals: gradient and equations
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CAIE P1 2021 June Q6
5 marks Standard +0.3
6 The equation of a curve is \(y = ( 2 k - 3 ) x ^ { 2 } - k x - ( k - 2 )\), where \(k\) is a constant. The line \(y = 3 x - 4\) is a tangent to the curve. Find the value of \(k\).
CAIE P1 2021 June Q3
6 marks Moderate -0.3
3 A line with equation \(y = m x - 6\) is a tangent to the curve with equation \(y = x ^ { 2 } - 4 x + 3\).
Find the possible values of the constant \(m\), and the corresponding coordinates of the points at which the line touches the curve.
CAIE P1 2023 November Q6
7 marks Standard +0.3
6 A line has equation \(y = 6 x - c\) and a curve has equation \(y = c x ^ { 2 } + 2 x - 3\), where \(c\) is a constant. The line is a tangent to the curve at point \(P\). Find the possible values of \(c\) and the corresponding coordinates of \(P\).
CAIE P1 2013 June Q7
9 marks Moderate -0.3
7 A curve has equation \(y = x ^ { 2 } - 4 x + 4\) and a line has equation \(y = m x\), where \(m\) is a constant.
  1. For the case where \(m = 1\), the curve and the line intersect at the points \(A\) and \(B\). Find the coordinates of the mid-point of \(A B\).
  2. Find the non-zero value of \(m\) for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.
AQA C1 2016 June Q6
8 marks Standard +0.3
6
  1. A curve has equation \(y = 8 - 4 x - 2 x ^ { 2 }\).
    1. Find the values of \(x\) where the curve crosses the \(x\)-axis, giving your answer in the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
    2. Sketch the curve, giving the value of the \(y\)-intercept.
  2. A line has equation \(y = k ( x + 4 )\), where \(k\) is a constant.
    1. Show that the \(x\)-coordinates of any points of intersection of the line with the curve \(y = 8 - 4 x - 2 x ^ { 2 }\) satisfy the equation $$2 x ^ { 2 } + ( k + 4 ) x + 4 ( k - 2 ) = 0$$
    2. Find the values of \(k\) for which the line is a tangent to the curve \(y = 8 - 4 x - 2 x ^ { 2 }\).
      [0pt] [3 marks]