Tangent meets curve/axis — further geometry

Find the tangent equation and then use it to find where it meets the x-axis, y-axis, or the curve again, or find midpoints/intersections involving the tangent line.

28 questions · Moderate -0.2

1.07m Tangents and normals: gradient and equations
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OCR C3 Q9
12 marks Standard +0.3
$$\text{f}(x) = e^{3x + 1} - 2, \quad x \in \mathbb{R}.$$
  1. State the range of f. [1]
The curve \(y = \text{f}(x)\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).
  1. Find the exact coordinates of \(P\) and \(Q\). [3]
  2. Show that the tangent to the curve at \(P\) has the equation $$y = 3ex + e - 2.$$ [4]
  3. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\). [4]
AQA AS Paper 1 2020 June Q8
8 marks Standard +0.3
  1. Find the equation of the tangent to the curve \(y = e^{4x}\) at the point \((a, e^{4a})\). [3 marks]
  2. Find the value of \(a\) for which this tangent passes through the origin. [2 marks]
  3. Hence, find the set of values of \(m\) for which the equation $$e^{4x} = mx$$ has no real solutions. [3 marks]
WJEC Unit 1 2022 June Q11
15 marks Standard +0.3
The diagram below shows a sketch of the curve \(y = f(x)\), where \(f(x) = 10x + 3x^2 - x^3\). The curve intersects the \(x\)-axis at the origin \(O\) and at the points \(A(-2, 0)\), \(B(5, 0)\). The tangent to the curve at the point \(C(2, 24)\) intersects the \(y\)-axis at the point \(D\). \includegraphics{figure_11}
  1. Find the coordinates of \(D\). [5]
  2. Find the area of the shaded region. [6]
  3. Determine the range of values of \(x\) for which \(f(x)\) is an increasing function. [4]