1.07m Tangents and normals: gradient and equations

873 questions

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Edexcel C12 2019 June Q10
9 marks Moderate -0.3
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } + 4 x + p y + 123 = 0$$ where \(p\) is a constant. Given that the point \(( 1,16 )\) lies on \(C\),
  1. find
    1. the value of \(p\),
    2. the coordinates of the centre of \(C\),
    3. the radius of \(C\).
  2. Find an equation of the tangent to \(C\) at the point ( 1,16 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-31_33_19_2668_1896}
Edexcel C12 2016 October Q13
13 marks Standard +0.3
13. The circle \(C\) has centre \(A ( 1 , - 3 )\) and passes through the point \(P ( 8 , - 2 )\).
  1. Find an equation for the circle \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(P\).
  2. Find an equation for \(l _ { 1 }\), giving your answer in the form \(y = m x + c\) The line \(l _ { 2 }\), with equation \(y = x + 6\), is the tangent to \(C\) at the point \(Q\).
  3. Find the coordinates of the point \(Q\).
Edexcel C12 2016 October Q14
11 marks Challenging +1.2
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-40_456_689_269_623} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = - x ^ { 2 } + 6 x - 8\). The normal to \(C\) at the point \(P ( 5 , - 3 )\) is the line \(l\), which is also shown in Figure 3.
  1. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The finite region \(R\), shown shaded in Figure 3, is bounded below by the line \(l\) and the curve \(C\), and is bounded above by the \(x\)-axis.
  2. Find the exact value of the area of \(R\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2017 October Q16
5 marks Moderate -0.3
  1. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 2 x - 5\), where \(a\) and \(b\) are constants The point \(P ( 1,4 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).
The tangent to \(y = \mathrm { f } ( x )\) at the point \(P\) has equation \(y = 12 x - 8\) Calculate the value of \(a\) and the value of \(b\).
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Q16
Edexcel C12 2018 October Q14
11 marks Standard +0.8
14. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 16 y + k = 0$$ where \(k\) is a constant.
  1. Find the coordinates of the centre of \(C\). Given that the radius of \(C\) is 10
  2. find the value of \(k\). The point \(A ( a , - 16 )\), where \(a > 0\), lies on the circle \(C\). The tangent to \(C\) at the point \(A\) crosses the \(x\)-axis at the point \(D\) and crosses the \(y\)-axis at the point \(E\).
  3. Find the exact area of triangle \(O D E\).
Edexcel C1 2005 January Q7
10 marks Moderate -0.8
7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .
  1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
  2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
  3. Find the value of \(k\).
Edexcel C1 2005 January Q9
11 marks Moderate -0.3
9. The gradient of the curve \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 } .$$ The point \(P ( 1,4 )\) lies on \(C\).
  1. Find an equation of the normal to \(C\) at \(P\).
  2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
  3. Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).
Edexcel C1 2006 January Q9
12 marks Easy -1.2
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{815e288c-0140-4c12-9e89-b0bb4fb1a8c1-12_812_1088_317_427}
\end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the \(x\)-axis at the points \(P , ( 1,0 )\) and \(Q\), as shown in Figure 2.
  1. Write down the \(x\)-coordinate of \(P\), and the \(x\)-coordinate of \(Q\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4\).
  3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point ( \(- 1,6\) ). The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point ( \(- 1,6\) ).
  4. Find the exact coordinates of \(R\).
Edexcel C1 2007 January Q7
9 marks Moderate -0.3
7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \neq 0\), and the point \(P ( 2,1 )\) lies on \(C\). Given that $$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 6 - \frac { 8 } { x ^ { 2 } } ,$$
  1. find \(\mathrm { f } ( x )\).
  2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.
Edexcel C1 2007 January Q8
11 marks Moderate -0.8
8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P ( 4,8 )\) lies on \(C\).
  3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3 y = x + 20 .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
  4. Find the length \(P Q\), giving your answer in a simplified surd form.
Edexcel C1 2008 January Q9
10 marks Moderate -0.3
9. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), and \(\mathrm { f } ^ { \prime } ( x ) = 4 x - 6 \sqrt { } x + \frac { 8 } { x ^ { 2 } }\). Given that the point \(P ( 4,1 )\) lies on \(C\),
  1. find \(\mathrm { f } ( x )\) and simplify your answer.
  2. Find an equation of the normal to \(C\) at the point \(P ( 4,1 )\).
Edexcel C1 2009 January Q11
13 marks Moderate -0.3
  1. The curve \(C\) has equation
$$y = 9 - 4 x - \frac { 8 } { x } , \quad x > 0$$ The point \(P\) on \(C\) has \(x\)-coordinate equal to 2 .
  1. Show that the equation of the tangent to \(C\) at the point \(P\) is \(y = 1 - 2 x\).
  2. Find an equation of the normal to \(C\) at the point \(P\). The tangent at \(P\) meets the \(x\)-axis at \(A\) and the normal at \(P\) meets the \(x\)-axis at \(B\).
  3. Find the area of triangle \(A P B\).
Edexcel C1 2010 January Q6
8 marks Moderate -0.8
6. The curve \(C\) has equation $$y = \frac { ( x + 3 ) ( x - 8 ) } { x } , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
  2. Find an equation of the tangent to \(C\) at the point where \(x = 2\)
Edexcel C1 2011 January Q11
12 marks Moderate -0.8
11. The curve \(C\) has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
  3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-15_113_129_2405_1816}
Edexcel C1 2012 January Q8
10 marks Moderate -0.8
8. The curve \(C _ { 1 }\) has equation $$y = x ^ { 2 } ( x + 2 )$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Sketch \(C _ { 1 }\), showing the coordinates of the points where \(C _ { 1 }\) meets the \(x\)-axis.
  3. Find the gradient of \(C _ { 1 }\) at each point where \(C _ { 1 }\) meets the \(x\)-axis. The curve \(C _ { 2 }\) has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where \(k\) is a constant and \(k > 2\)
  4. Sketch \(C _ { 2 }\), showing the coordinates of the points where \(C _ { 2 }\) meets the \(x\) and \(y\) axes.
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 \(B\).
Edexcel C1 2013 January Q11
12 marks Moderate -0.8
11. The curve \(C\) has equation $$y = 2 x - 8 \sqrt { } x + 5 , \quad x \geqslant 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form. The point \(P\) on \(C\) has \(x\)-coordinate equal to \(\frac { 1 } { 4 }\)
  2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants. The tangent to \(C\) at the point \(Q\) is parallel to the line with equation \(2 x - 3 y + 18 = 0\)
  3. Find the coordinates of \(Q\).
Edexcel C1 2014 January Q10
10 marks Moderate -0.3
10. The curve \(C\) has equation \(y = x ^ { 3 } - 2 x ^ { 2 } - x + 3\) The point \(P\), which lies on \(C\), has coordinates \(( 2,1 )\).
  1. Show that an equation of the tangent to \(C\) at the point \(P\) is \(y = 3 x - 5\) The point \(Q\) also lies on \(C\).
    Given that the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\),
  2. find the coordinates of the point \(Q\).
Edexcel C1 2005 June Q10
11 marks Moderate -0.3
10. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3\). The point \(P\) has coordinates \(( 3,0 )\).
  1. Show that \(P\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  3. Find the coordinates of \(Q\).
Edexcel C1 2006 June Q10
10 marks Moderate -0.3
10. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \neq 0\), passes through the point ( \(3,7 \frac { 1 } { 2 }\) ). Given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 3 } { x ^ { 2 } }\),
  1. find \(\mathrm { f } ( x )\).
  2. Verify that \(f ( - 2 ) = 5\).
  3. Find an equation for the tangent to \(C\) at the point ( \(- 2,5\) ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2007 June Q10
13 marks Moderate -0.3
10. The curve \(C\) has equation \(y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 2 respectively.
  1. Show that the length of \(P Q\) is \(\sqrt { 170 }\).
  2. Show that the tangents to \(C\) at \(P\) and \(Q\) are parallel.
  3. Find an equation for the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. \(\_\_\_\_\)
Edexcel C1 2009 June Q11
11 marks Standard +0.8
11. The curve \(C\) has equation $$y = x ^ { 3 } - 2 x ^ { 2 } - x + 9 , \quad x > 0$$ The point \(P\) has coordinates (2, 7).
  1. Show that \(P\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The point \(Q\) also lies on \(C\).
    Given that the tangent to \(C\) at \(Q\) is perpendicular to the tangent to \(C\) at \(P\),
  3. show that the \(x\)-coordinate of \(Q\) is \(\frac { 1 } { 3 } ( 2 + \sqrt { 6 } )\).
Edexcel C1 2010 June Q11
9 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\), where
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x - \frac { 5 } { \sqrt { } x } - 2$$ Given that the point \(P ( 4,5 )\) lies on \(C\), find
  1. \(\mathrm { f } ( x )\),
  2. an equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2011 June Q10
14 marks Moderate -0.3
10. The curve \(C\) has equation $$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$
  1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
  3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
  4. Find the \(x\)-coordinate of \(B\).
Edexcel C1 2012 June Q7
8 marks Moderate -0.3
7. The point \(P ( 4 , - 1 )\) lies on the curve \(C\) with equation \(y = \mathrm { f } ( x ) , x > 0\), and $$f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - \frac { 6 } { \sqrt { } x } + 3$$
  1. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.
  2. Find \(\mathrm { f } ( x )\).