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Prove that the equation of the tangent and normal to the ellipse (x^2/a^2) + (y^2/b^2) = 1 at the point P (acosu , bsinu) are respectively
bxcosu + aysinu = ab and
axsinu + bycosu = (a^2 - b^2)sinucosu.
Prove that the equation normal to the rectangular hyperbola xy = c^2 at the point P(ct, c/t) is t^3*x - ty = c(t^4 - 1)
Write down the standard form of the equation of a parabola in a given directrix and vertex at (0, 0)
(a) x = -4
(b) y = 2
(c) y = - 3
A particle P moves on the curve with polar equation r = 1/ (2 - sinx) . Given that at any instant t, during the motion, r^2 (dx/dt) = 4,
(i) write an expression for r(dx/dt) in terms of x.
(ii) Show that dr/dt = 4cosx and 1/3 <=r<=1.
(iii) Find the speed of P when x = 0.
(iv) Prove that the force acting on P is directed towards the pole.

Find the equation of the tangent to the plane f(x,y) at the point (-1,2)

F(X,y )= 2x^2 -e^2x-3y-8


Shade the region of the xy-plane for which

a) (𝑥2+𝑦2−16)(𝑥2−4)≤0

b) (𝑦−𝑥)(𝑦2+𝑥3)>0


Find the rectangular coordinates of the point whose spherical coordinates are
1) (5, π/2,π/2)
2) (4, π/3, 2π/3)
3) (0,π/11,π/5)
4) (2, 5π/3, 3π/4)
Find the spherical coordinates of the point whose coordinates are
1) (1, 1, √6)
2) (-2, 2√3, 4)
3) (-√3, 1, -2)
4) (4,-4√3, 6)
Change the following from cylindrical coordinates to rectangular coordinates
1) (5, π/6, 3)
2) (6, π/3, -5)
Show that. The curvature and torsion of the straight line is zero.
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