- Given that \(y = \operatorname { artanh } \frac { x } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
- \hspace{0pt} [In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.]
The points \(P ( 3 \cos \alpha , 2 \sin \alpha )\) and \(Q ( 3 \cos \beta , 2 \sin \beta )\), where \(\alpha \neq \beta\), lie on the ellipse with equation
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
- Show the equation of the chord \(P Q\) is
$$\frac { x } { 3 } \cos \frac { ( \alpha + \beta ) } { 2 } + \frac { y } { 2 } \sin \frac { ( \alpha + \beta ) } { 2 } = \cos \frac { ( \alpha - \beta ) } { 2 }$$
- Write down the coordinates of the mid-point of \(P Q\).
Given that the gradient, \(m\), of the chord \(P Q\) is a constant,
- show that the centre of the chord lies on a line
$$y = - k x$$
expressing \(k\) in terms of \(m\).