Hyperbola tangent/normal equation derivation

A question is this type if and only if it asks to derive the equation of a tangent or normal to a hyperbola at a parametric point using calculus.

4 questions · Challenging +1.5

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Edexcel F3 2017 June Q6
9 marks Challenging +1.8
  1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
    and the ellipse \(E\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
    where \(a > b > 0\)
    The line \(l\) is a tangent to hyperbola \(H\) at the point \(P ( a \sec \theta , b \tan \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)
    1. Using calculus, show that an equation for \(l\) is
    $$b x \sec \theta - a y \tan \theta = a b$$ Given that the point \(F\) is the focus of ellipse \(E\) for which \(x > 0\) and that the line \(l\) passes through \(F\),
  2. show that \(l\) is parallel to the line \(y = x\)
Edexcel FP3 Q8
12 marks Challenging +1.8
8. The point \(\mathrm { P } ( 5 \sec \mathrm { u } , 3 \tan \mathrm { u } )\) lies on the hyperbola H with equation \(\frac { \mathrm { x } ^ { 2 } } { 25 } - \frac { \mathrm { y } ^ { 2 } } { 9 } = 1\). The tangent to \(H\) at \(P\) intersects the asymptote of \(H\) with equation \(y = \frac { 3 } { 5 } x\) at the point \(R\) and the asymptote with equation \(\mathrm { y } = - \frac { 3 } { 5 } \mathrm { x }\) at the point S .
  1. Use differentiation to show that an equation of the tangent to H at P is $$3 x = 5 y \sin u + 15 \cos u$$
  2. Prove that P is the mid-point of RS.
Edexcel FP3 2016 June Q5
11 marks Challenging +1.2
5. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The point \(P ( 4 \sec \theta , 3 \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\), lies on \(H\).
  1. Show that an equation of the normal to \(H\) at the point \(P\) is $$3 y + 4 x \sin \theta = 25 \tan \theta$$ The line \(l\) is the directrix of \(H\) for which \(x > 0\)
    The normal to \(H\) at \(P\) crosses the line \(l\) at the point \(Q\). Given that \(\theta = \frac { \pi } { 4 }\)
  2. find the \(y\) coordinate of \(Q\), giving your answer in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers to be found.
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Edexcel FP3 Q16
14 marks Challenging +1.2
16. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\).
  1. Show that an equation of the normal to \(C\) at the point \(P ( a \sec t , b \tan t )\) is $$a x \sin t + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan t .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and \(S\) is a focus of \(C\). Given that the eccentricity of \(C\) is \(\frac { 3 } { 2 }\), and that \(O A = 3 O S\), where \(O\) is the origin,
  2. determine the possible values of \(t\), for \(0 \leq t < 2 \pi\).
    [0pt] [P5 June 2003 Qn 1]