Tangency condition for two curves

A question is this type if and only if it asks to find parameter values for which two curves meet at a single point or are tangent to each other.

1 questions · Standard +0.3

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Edexcel P1 2024 June Q4
6 marks Standard +0.3
  1. The curve \(C _ { 1 }\) has equation
$$y = x ^ { 2 } + k x - 9$$ and the curve \(C _ { 2 }\) has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where \(k\) is a constant.
Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at a single point \(P\)
  1. show that $$k ^ { 2 } + 26 k + 169 = 0$$
  2. Hence find the coordinates of \(P\)