OCR C4 (Core Mathematics 4)

3 The line \(L _ { 1 }\) passes through the points \(( 2 , - 3,1 )\) and \(( - 1 , - 2 , - 4 )\). The line \(L _ { 2 }\) passes through the point \(( 3,2 , - 9 )\) and is parallel to the vector \(4 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\).

1. Find an equation for \(L _ { 1 }\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
2. Prove that \(L _ { 1 }\) and \(L _ { 2 }\) are skew.

7 A curve is given parametrically by the equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
2. Show that the equation of the tangent at the point \(P \left( 4 , - \frac { 1 } { 2 } \right)\) is $$x - 16 y = 12$$
3. Find the value of the parameter at the point where the tangent at \(P\) meets the curve again. June 2005