Show line is tangent, verify

A question is this type if and only if a specific line and curve are both fully given (no unknown constants) and the task is to verify or show that the line is a tangent to the curve by demonstrating the discriminant equals zero or a repeated root exists.

10 questions · Moderate -0.5

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CAIE P1 2008 June Q4
7 marks Moderate -0.8
4 The equation of a curve \(C\) is \(y = 2 x ^ { 2 } - 8 x + 9\) and the equation of a line \(L\) is \(x + y = 3\).
  1. Find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
  2. Show that one of these points is also the stationary point of \(C\).
Edexcel P1 2023 June Q10
10 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-28_903_1010_219_539} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the quadratic curve \(C\) with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line \(l _ { 1 }\) also shown in Figure 5,
  • has gradient \(\frac { 1 } { 2 }\)
  • intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
  • find, in terms of \(b\), an equation for \(l _ { 2 }\) Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)
  • show that another equation for \(l _ { 2 }\) is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$
  • Hence, or otherwise, find the value of \(b\)
Edexcel C1 2012 January Q10
11 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff1cdb91-0286-4bc8-9e67-451500b2bf74-14_769_935_285_411} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$y = 2 - \frac { 1 } { x } , \quad x \neq 0$$ The curve crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\).
  2. Show that the equation of the normal to \(C\) at \(A\) can be written as $$2 x + 8 y - 1 = 0$$ The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 2 .
  3. Find the coordinates of \(B\).
OCR MEI C1 2008 June Q10
12 marks Moderate -0.8
10
  1. Express \(x ^ { 2 } - 6 x + 2\) in the form \(( x - a ) ^ { 2 } - b\).
  2. State the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 2\).
  3. Sketch the graph of \(y = x ^ { 2 } - 6 x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis.
  4. Solve the simultaneous equations \(y = x ^ { 2 } - 6 x + 2\) and \(y = 2 x - 14\). Hence show that the line \(y = 2 x - 14\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 2\).
OCR C1 Q7
9 marks Moderate -0.8
7. (i) Calculate the discriminant of \(x ^ { 2 } - 6 x + 12\).
(ii) State the number of real roots of the equation \(x ^ { 2 } - 6 x + 12 = 0\) and hence, explain why \(x ^ { 2 } - 6 x + 12\) is always positive.
(iii) Show that the line \(y = 8 - 2 x\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 12\).
OCR MEI C1 Q1
12 marks Moderate -0.3
1
  1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
  2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
  3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes. [4] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{973ad9eb-33f2-432e-9449-e54c1728008b-1_1292_1401_887_359} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\).
  4. Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
  5. Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
  6. Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation. [5]
  7. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
  8. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
  9. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
    (A) write down the equation of the line of symmetry,
    (B) write down the minimum \(y\)-value on the curve.
OCR MEI C1 2013 January Q11
12 marks Moderate -0.3
11
  1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
  2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
  3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes.
AQA C1 2006 January Q3
9 marks Moderate -0.8
3
    1. Express \(x ^ { 2 } - 4 x + 9\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Hence, or otherwise, state the coordinates of the minimum point of the curve with equation \(y = x ^ { 2 } - 4 x + 9\).
  2. The line \(L\) has equation \(y + 2 x = 12\) and the curve \(C\) has equation \(y = x ^ { 2 } - 4 x + 9\).
    1. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation $$x ^ { 2 } - 2 x - 3 = 0$$
    2. Hence find the coordinates of the points of intersection of \(L\) and \(C\).
Edexcel C1 Q9
11 marks Moderate -0.8
9. The curve \(C\) has the equation \(y = x ^ { 2 } + 2 x + 4\).
  1. Express \(x ^ { 2 } + 2 x + 4\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point of \(C\). The straight line \(l\) has the equation \(x + y = 8\).
  2. Sketch \(l\) and \(C\) on the same set of axes.
  3. Find the coordinates of the points where \(I\) and \(C\) intersect.
AQA C1 2007 June Q3
12 marks Moderate -0.8
3
    1. Express \(x ^ { 2 } + 10 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Write down the coordinates of the vertex (minimum point) of the curve with equation \(y = x ^ { 2 } + 10 x + 19\).
    3. Write down the equation of the line of symmetry of the curve \(y = x ^ { 2 } + 10 x + 19\).
    4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 10 x + 19\).
  2. Determine the coordinates of the points of intersection of the line \(y = x + 11\) and the curve \(y = x ^ { 2 } + 10 x + 19\).