Show/verify a given line is a tangent

Prove or verify that a given line is tangent to a curve, typically by showing it touches the curve at exactly one point or that the gradient matches.

4 questions · Standard +0.0

1.07m Tangents and normals: gradient and equations

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Pre-U Pre-U 9794/1 Specimen Q3

5 marks Moderate -0.8

3 Show that the equation of the tangent to the curve \(y = \ln \left( x ^ { 2 } + 3 \right)\) at the point \(( 1 , \ln 4 )\) is $$2 y - x = \ln ( 16 ) - 1$$

CAIE P1 2011 November Q7

9 marks Moderate -0.3

A straight line passes through the point \((2, 0)\) and has gradient \(m\). Write down the equation of the line. [1]
Find the two values of \(m\) for which the line is a tangent to the curve \(y = x^2 - 4x + 5\). For each value of \(m\), find the coordinates of the point where the line touches the curve. [6]
Express \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\) and hence, or otherwise, write down the coordinates of the minimum point on the curve. [2]

Edexcel P1 2018 Specimen Q4

5 marks Standard +0.3

The straight line with equation \(y = 4x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = 2x^2 + 8x + 3\) Calculate the value of \(c\) [5]

AQA AS Paper 1 2022 June Q10

9 marks Standard +0.8

Curve \(C\) has equation \(y = \frac{\sqrt{2}}{x^2}\)

Find an equation of the tangent to \(C\) at the point \(\left(2, \frac{\sqrt{2}}{4}\right)\) [4 marks]
Show that the tangent to \(C\) at the point \(\left(2, \frac{\sqrt{2}}{4}\right)\) is also a normal to the curve at a different point. \includegraphics{figure_10} [5 marks]