Normal to curve at given point

Find equation of normal line (perpendicular to tangent) to a curve at a specified point

3 questions · Standard +0.3

1.07m Tangents and normals: gradient and equations
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CAIE P1 2008 November Q8
9 marks Standard +0.3
8 The equation of a curve is \(y = 5 - \frac { 8 } { x }\).
  1. Show that the equation of the normal to the curve at the point \(P ( 2,1 )\) is \(2 y + x = 4\). This normal meets the curve again at the point \(Q\).
  2. Find the coordinates of \(Q\).
  3. Find the length of \(P Q\).
CAIE P1 2010 November Q10
10 marks Standard +0.3
10 The equation of a curve is \(y = 3 + 4 x - x ^ { 2 }\).
  1. Show that the equation of the normal to the curve at the point \(( 3,6 )\) is \(2 y = x + 9\).
  2. Given that the normal meets the coordinate axes at points \(A\) and \(B\), find the coordinates of the mid-point of \(A B\).
  3. Find the coordinates of the point at which the normal meets the curve again.
Edexcel C1 2013 June Q11
11 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-15_592_1394_274_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(H\) with equation \(y = \frac { 3 } { x } + 4 , x \neq 0\).
  1. Give the coordinates of the point where \(H\) crosses the \(x\)-axis.
  2. Give the equations of the asymptotes to \(H\).
  3. Find an equation for the normal to \(H\) at the point \(P ( - 3,3 )\). This normal crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  4. Find the length of the line segment \(A B\). Give your answer as a surd.