PURE MATHEMATICS 212 Multivariable Calculus ASSIGNMENT 2 1. [2 marks] Name and sketch the surface: z = 4x2 + y2 + 8x ?? 2y 2. [4 marks] (a) Find a parametric equation of the curve of intersection of...

1) Given the surface, ? = 4?2 + ?2 + 8? − 2? below is the sketch of it.
It can also be written as ? + 5 = 4(? + 1)2 + (? − 1)2
“An Elliptical Paraboloid”
2) A)
Curve of intersection of ? = 3 − ?2 − ?2 & ? = 2? is found by equating
those two equations i.e. 3 − ?2 − ?2 = 2? which gives
?2 + ?2 + 2? = 3
Adding 1 on both sides and rearranging gives,
(? − 0)2 + (? + 1)2 = 22 , which is a circle centred at (0,-1) and radius 2.
Since ? = 2?, centre of the circle is (0, −1, −2) & ?????? = 2.
So, writing in parametric equation it will be
"? = ? ???(?) , ? = −? + ? ???(?) , ? = −? + ? ???(?) , ? ∈ (?, ??)"
B) Orthogonal projection of this curve on XY plane is nothing but,
(? − ?)? + (? + ?)? = ??, a circle centred at (0,-1) and radius 2.
3) A) Given ? = (3 sin(??))? + (3 cos(??))?
This implies, ? = (3 sin(??)) & ? = (3 cos(??)), clearly this parametric
equation corresponds to a circle and its equation in Cartesian co-ordinates
is ?? + ?? = ?, a ??????
B) Given ? = −2? + ?? + (?2 − 1)?
This implies ? = −2 & ? = ? & ? = ?2 − 1 = ?2 − 1, thus the equation
in Cartesian co-ordinates is ? = ?? − ? & ? = −?, a parabola...