Summary MAT2615 ASSIGNMENT 2 2024

MAT2615
ASSINGMENT 2
2024

, QUESTION 1




Solution:



a).



f(x, y) = c



f(x, y) = c
1 + x2 + y2 = c ∴ x = 1 and y = 1

1 + (1)2 + (1)2 = c
c=3




f(x, y) = 3

1 + x2 + y2 = 3

x2 + y2 = 3 − 1

x2 + y2 = 2
2
x 2 + y 2 = (√2) a circle with center at origin and radius √2




2
Equation of Curve C: x 2 + y 2 = (√2)

, b).



⃗ = vector that is perpendicular to C at (1,1)
Let: n



⃗ = 〈fx (1,1), fy (1,1)〉
n



f(x, y) = 1 + x 2 + y 2



fx (x, y) = 2x and fy (x, y) = 2y



fx (1,1) = 2(1) and fy (1,1) = 2(1)

fx (1,1) = 2 and fy (1,1) = 2




⃗ = (fx (1,1), fy (1,1))
n

⃗ = (2,2)
n



c).



Equation of line: (x, y) = (xo , y0 ) + tn
⃗


(x, y) = (xo , y0 ) + tn
⃗ ∴ (xo , y0 ) = (1,1)
(x, y) = (1,1) + t(2,2)
(x, y) = (1 + 2t, 1 + 2t)



x = 1 + 2t and y = 1 + 2t
x−1 y−1
t= and t =
2 2


x−1 y−1
=
2 2