Analytic Geometry Answers

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

3x + 8y = 4

(8, -9)

a. Find the equation of a line passing through the point given perpendicular to the line given.

b. Find the distance between the point and the line.

c. Find the equation of a line passing through the point (4, 3) parallel to the line given and its distance.

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)