Hyperbola focus-directrix properties

1. A hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 5 } = 1 \quad \text { where } a \text { is a positive constant }$$ The line with equation \(x = \frac { 4 } { 3 }\) is a directrix of \(H\)

1. Write down an equation of the other directrix.
2. Determine
   1. the value of \(a\)
   2. the coordinates of each of the foci of \(H\)

2. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
The hyperbola \(H\) has eccentricity \(\frac { \sqrt { 21 } } { 4 }\) and passes through the point (12, 5).
Find

1. the value of \(a\) and the value of \(b\),
2. the coordinates of the foci of \(H\).

8. The hyperbola \(H\) has equation $$4 x ^ { 2 } - y ^ { 2 } = 4$$

1. Write down the equations of the asymptotes of \(H\).
2. Find the coordinates of the foci of \(H\). The point \(P ( \sec \theta , 2 \tan \theta )\) lies on \(H\).
3. Using calculus, show that the equation of the tangent to \(H\) at the point \(P\) is $$y \tan \theta = 2 x \sec \theta - 2$$ The point \(V ( - 1,0 )\) and the point \(W ( 1,0 )\) both lie on \(H\).
   The point \(Q ( \sec \theta , - 2 \tan \theta )\) also lies on \(H\).
   Given that \(P , Q , V\) and \(W\) are distinct points on \(H\) and that the lines \(V P\) and \(W Q\) intersect at the point \(S\),
4. show that, as \(\theta\) varies, \(S\) lies on an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are integers to be found.
   
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1. The hyperbola \(H\) has

• foci with coordinates \(\left( \pm \frac { 13 } { 2 } , 0 \right)\)
• directrices with equations \(x = \pm \frac { 72 } { 13 }\)
• eccentricity e

Determine

1. the value of \(e\)
2. an equation for \(H\), giving your answer in the form \(p x ^ { 2 } - q y ^ { 2 } = r\), where \(p , q\) and \(r\) are integers.

7. A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1$$ where \(a\) is a positive constant.
The eccentricity of \(H\) is \(e\).

1. Determine an expression for \(e ^ { 2 }\) in terms of \(a\). The line \(l\) is the directrix of \(H\) for which \(x > 0\)
   The points \(A\) and \(A ^ { \prime }\) are the points of intersection of \(l\) with the asymptotes of \(H\).
2. Determine, in terms of \(e\), the length of the line segment \(A A ^ { \prime }\). The point \(F\) is the focus of \(H\) for which \(x < 0\)
   Given that the area of triangle \(A F A ^ { \prime }\) is \(\frac { 164 } { 3 }\)
3. show that \(a\) is a solution of the equation $$30 a ^ { 3 } - 164 a ^ { 2 } + 375 a - 4100 = 0$$
4. Hence, using algebra and making your reasoning clear, show that the only possible value of \(a\) is \(\frac { 20 } { 3 }\)

1. The hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ Find

1. the coordinates of the foci of \(H\),
2. the equations of the directrices of \(H\).

1. The hyperbola \(H\) has foci at \(( 5,0 )\) and \(( - 5,0 )\) and directrices with equations \(x = \frac { 9 } { 5 }\) and \(x = - \frac { 9 } { 5 }\).

Find a cartesian equation for \(H\).

1. A hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1 , \quad \text { where } a \text { is a positive constant. }$$ The foci of \(H\) are at the points with coordinates \(( 13,0 )\) and \(( - 13,0 )\).
Find

1. the value of the constant \(a\),
2. the equations of the directrices of \(H\).

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$$ Find

1. the coordinates of the foci of \(E\),
2. the equations of the directrices of \(E\).

1. The parabola \(C\) has equation

$$y ^ { 2 } = 32 x$$ and the hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 9 } = 1$$

1. Write down the equations of the asymptotes of \(H\). The line \(l _ { 1 }\) is normal to \(C\) and parallel to the asymptote of \(H\) with positive gradient. The line \(l _ { 2 }\) is normal to \(C\) and parallel to the asymptote of \(H\) with negative gradient.
2. Determine
   1. an equation for \(l _ { 1 }\)
   2. an equation for \(l _ { 2 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet \(H\) at the points \(P\) and \(Q\) respectively.
3. Find the area of the triangle \(O P Q\), where \(O\) is the origin.

1. An ellipse has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1\) and eccentricity \(e _ { 1 }\) A hyperbola has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) and eccentricity \(e _ { 2 }\)

Given that \(e _ { 1 } \times e _ { 2 } = 1\)

1. show that \(a ^ { 2 } = 3 b ^ { 2 }\) Given also that the coordinates of the foci of the ellipse are the same as the coordinates of the foci of the hyperbola,
2. determine the equation of the hyperbola.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$ The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
Given that

• the eccentricity of \(H\) is the reciprocal of the eccentricity of \(E\)
• the coordinates of the foci of \(H\) are the same as the coordinates of the foci of \(E\) determine
  1. the value of \(a\)
  2. the value of \(b\)

5

1. Write down the equations of the asymptotes of \(H\). 5
2. Sketch the hyperbola \(H\) on the axes below, indicating the coordinates of any points of intersection with the coordinate axes. The asymptotes have already been drawn.
   [0pt] [2 marks]
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3. The finite region bounded by \(H\), the positive \(x\)-axis, the positive \(y\)-axis and the line \(y = a\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Show that the volume of the solid generated is \(m a ^ { 3 }\), where \(m = 3.40\) correct to three significant figures.

3 Find the equations of the asymptotes of the curve \(x ^ { 2 } - 3 y ^ { 2 } = 1\) Circle your answer.
[0pt] [1 mark] $$y = \pm 3 x \quad y = \pm \frac { 1 } { 3 } x \quad y = \pm \sqrt { 3 } x \quad y = \pm \frac { 1 } { \sqrt { 3 } } x$$ Turn over for the next question
\(\mathbf { 4 } \quad \mathbf { A } = \left[ \begin{array} { l l } 1 & 2 \\ 1 & k \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { c c } - 1 & 0 \\ 0 & 1 \end{array} \right]\)