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Problem 1 (25 Points)
Consider the lines L1L_1L1 and L2L_2L2 with parametric equations
L1:x=1+3t,y=2−t,z=−3+4t.L_1 : \quad x = 1 + 3t, \quad y = 2 – t, \quad z = -3 + 4t.
L2:x=1−s,y=−2+s,z=4+2s.L_2 : \quad x = 1 – s, \quad y = -2 + s, \quad z = 4 + 2s.
a) Are these lines L1L_1L1 and L2L_2L2 parallel to each other? Justify your answer.
b) Do the lines L1L_1L1 and L2L_2L2 intersect at some point? Justify your answer.
Problem 2 (25 Points)
Given the following implicit equation
379+98x+7×2−32y+4y2=0379 + 98x + 7x^2 – 32y + 4y^2 = 0
a) Find the center and all four vertices of the ellipse described by the equation after writing it in standard form.
b) Parameterise all points on the ellipse using a minimal number of functions y=f(x)y = f(x).
c) The equation is fulfilled for points (x,y)(x, y)(x,y) on the ellipse. What are the intervals XXX and YYY for admissible values x∈Xx \in X and y∈Yy \in Y?
Problem 3 (25 Points)
Given
L1:x−4−2=y+18=z−32,andL2:x=−t, y=−2+3t, z=5t.L_1 : \frac{x – 4}{-2} = \frac{y + 1}{8} = \frac{z – 3}{2}, \quad \text{and} \quad L_2 : x = -t, \; y = -2 + 3t, \; z = 5t.
a) Find the smallest angle between the lines L1L_1L1 and L2L_2L2.
b) Find the shortest distance from the point P=(2,7,5)P = (2,7,5) on the line L1L_1L1 to the line L2L_2L2.
Problem 4 (25 Points)
Let TTT be the triangle with vertices A,B,A, B,A,B, and CCC given by
A=(2,8,−2),B=(2,3,1),C=(1,0,5).A = (2,8,-2), \quad B = (2,3,1), \quad C = (1,0,5).
a) Calculate the area of the triangle TTT.
b) Find the equation of the plane containing the points A,B,A, B,A,B, and CCC.
c) What is the distance of this plane from the origin.
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