Analytic Geometry Answers

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).

1.Convert the polar coordinates (-8, ) into rectangular coordinates.

2.Convert the rectangular coordinates (3, -3) into polar coordinates with r > 0 and 0 ≤ θ < 2π.

3.Convert the rectangular equation x2 + y2 = 100 into a polar equation that expresses r in terms of θ.

4.Convert the polar equation 4r cos θ + r sin θ = 8 into a rectangular equation that expresses y in terms of x.

Plot the following points:

P1 (-3, 135°)

P2 (2, )

P3 (4, 405°)

Graph

Sketch the graph of r = 3 − 2cosθ.

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.

Find the equation of the circle(x - 5)² + (y + 3)² = 25  translated through a vector

(-2,-2). Also Sketch it.

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)