Analytic Geometry Answers

Does there pass a plane through the lines (x+4)/3 =y/2 =(z−1)/3
and x/2 =(y−1)/1 =(z+1)/1?Justify.

What is the new equation of the conic
x^2 +y^2 +4x−2y+3 = 0, when
i) the origin is shifted at (2,−1), followed by a rotation of axes through 45^◦
?
ii) the axes are rotated through 45^◦
, followed by the shifting of the origin at
(2,−1)?
Are the equations in i) and ii) above the same? Why?

What is the new equation of the conic
x^2 +y^2 +4x−2y+3 = 0, when
i) the origin is shifted at (2,−1), followed by a rotation of axes through 45^◦
?
ii) the axes are rotated through 45^◦
, followed by the shifting of the origin at
(2,−1)?

Find the equation of the plane passing through the line of intersection of the planes
2x+3y+z = 4 and x+y+z = 2, and which is perpendicular to the plane
2x+3y−z = 3.

If the equation of a parabola with the focus at (3,−4) and the directrix x+y = 2 is x^2 +y^2 −2xy−8x+20y+c = 0, then what is the value of c?

QUESTION 8 8.1 Let L1 and L2 be lines defined by x = w0 + su, s ∈ R and y = w1 + tv, t ∈ R, respectively. Show that L1and L2 are parallel if and only if u = kv for some k ∈ R.

Suppose u and v are nonzero vectors in 3-space, where u=(u1,u2, u3) and v=(v1,v2,v3). Prove that u × v is perpendicular to both u and v by making use of the dot product

3x - 4y + 3 = 0
6x - 8y +7 = 0
a. Find the distance between the two lines.
b. Find the equation of the perpendicular line passing through. (-6,4)
c. Determine the distance of 2 given equations to point (-6.4)

8.1 Let L1 and L2 be lines defined by
x = w0 + su; s 2 R
and y = w1 + tv; t 2 R; respectively:
Show that L1and L2 are parallel if and only if u = kv for some k 2 R:

The Polar equation r = theta represent a conic