Hyperbola locus problems

4. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l\) is a normal to \(H\) at the point \(P ( a \sec \theta , b \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\), and the point \(M\) is the midpoint of \(P Q\).
2. Find the coordinates of \(M\).
3. Hence find the cartesian equation of the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

8. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\). The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).

1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$

1. The hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$

1. Use calculus to show that the equation of the tangent to \(H\) at the point \(( a \cosh \theta , b \sinh \theta )\) may be written in the form $$x b \cosh \theta - y a \sinh \theta = a b$$ The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(( a \cosh \theta , b \sinh \theta ) , \theta \neq 0\).
   Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(P\),
2. find, in terms of \(a\) and \(\theta\), the coordinates of \(P\). The line \(l _ { 2 }\) is the tangent to \(H\) at the point ( \(a , 0\) ).
   Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(Q\),
3. find, in terms of \(a , b\) and \(\theta\), the coordinates of \(Q\).
4. Show that, as \(\theta\) varies, the locus of the mid-point of \(P Q\) has equation $$x \left( 4 y ^ { 2 } + b ^ { 2 } \right) = a b ^ { 2 }$$

9. 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 \(P ( a \sec \theta , b \tan \theta )\) is $$b y + a x \sin \theta = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(A B\) is \(M\).
2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies.
   (Total 13 marks)

1. The hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \cosh \theta , 3 \sinh \theta )\).
The line \(l _ { 1 }\) meets the \(x\)-axis at the point \(A\).
The line \(l _ { 2 }\) is the tangent to \(H\) at the point \(( 4,0 )\).
The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(B\) and the midpoint of \(A B\) is the point \(M\).

1. Show that, as \(\theta\) varies, a Cartesian equation for the locus of \(M\) is $$y ^ { 2 } = \frac { 9 ( 4 - x ) } { 4 x } \quad p < x < q$$ where \(p\) and \(q\) are values to be determined. Let \(S\) be the focus of \(H\) that lies on the positive \(x\)-axis.
2. Show that the distance from \(M\) to \(S\) is greater than 1

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\).

The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).

1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$