Analytic Geometry Answers

Known vectors:

a = (α, 0, 1), b = (1, β, 1), c = (1, 1, γ)

Determine the values of α, β,γ when the three vectors are orthogonal to each other.

Known vectors:

⃗a = (1, 0, 1) , ⃗b = (0, 1, -1) , ⃗c = (0, 0, 1)

Find the angle between:

1. a and b

2. a and c

3. b and c

Find the equations of the tangents and normal to the following curves:

1. y2 + 8x = 0, parallel to x + y + 4 = 0.
2. x2 = 3y, perpendicular to x – 2y + 7 = 0.
3. x2 + 9y2 = 25, parallel to 4x + 9y + 30 = 0.
4. 25x2 + 4y2 = 100, perpendicular to 8x – 15y + 4 = 0.
5. x2 – y2 = 15, parallel to 4x – y + 20 = 0.

Use the information provided to write the standard form equation of ellipse. Then sketch

the graph.

Foci: (-10, 16), (-10,-8)

Endpoints of major axis: (-10, 17),(-10,-9)

A semi-elliptical archway over a one-way road has a height of 9 ft and width of 36 ft. Your van has a width of 12 ft and height of 8 ft. Will your van be able to clear the opening of the archway?

Find the equations of the tangent and normal to each of the following conics, the lengths of the subtangent and subnormal, then trace the curve showing these lines.

1. y2 = -9 (x – 3) at (2, 3).
2. 4x2 + 3y2 + 16x + 36y – 68 = 0 at (4, -2)
3. 9x2 – 4y2 – 108x – 56y + 128 = 0 at (0, 2).
4. x2 + y2 – 8x – 6y + 15 = 0 at (5, 0).

Identify the center, vertices, and foci. Then sketch the graph.

((x + 3) ^ 2)/9 + (y ^ 2)/25 = 1

Find the equations of the tangent and normal to each of the following conics, the lengths of the subtangent and subnormal, then trace the curve showing these lines.

y = x2 – 6x + 4 at (4, -4)

find the center,foci,vertices, endpoints of conjugate axis.determine the equation of the asymptotes and sketsch the graph

1. (y+6)²/25-(x-4)²/39=1

2. 9x²+126x-16y²-96y+153=0

Identify the vertex, focus, axis of symmetry, directrix, direction of opening, length of the latus rectum, and the x- and y-intercepts of: 2y ^ 2 + x + 20y + 51 = 0