The slopes are m1 = 2 and m2 = -1/2. The angle θ between the lines is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)|, which results in 90 degrees.

The slopes are m1 = 3 and m2 = -1/3. The angle θ = tan⁻¹(|(m1 - m2) / (1 + m1*m2)|) = tan⁻¹(10/8) = 90 degrees.

The axis of symmetry for a parabola in vertex form y = a(x - h)^2 + k is x = h. Here, h = 2.

The axis of symmetry for the parabola y^2 = 4px is the x-axis.

For the equation y^2 = 4px, p = 5. The directrix is given by x = -p, which is x = -5.

Using the distance formula: d = √[(3 - 0)² + (4 - 0)²] = √[9 + 16] = √25 = 5.

Using the distance formula: d = √((8 - 0)² + (6 - 0)²) = √(64 + 36) = √100 = 10.

Using the distance formula: d = √[(3 - 0)² + (4 - 0)²] = √[9 + 16] = √25 = 5.

Using the distance formula: d = √[(6 - 2)² + (7 - 3)²] = √[16 + 16] = √32 = 4√2 ≈ 5.66.

Using the distance formula: d = √[(3 - 3)² + (-2 - 2)²] = √[0 + 16] = √16 = 4.

Using the distance formula: d = √[(3 - 3)² + (1 - 7)²] = √[0 + 36] = √36 = 6.

Using the distance formula: d = √[(1 - 5)² + (1 - 5)²] = √[16 + 16] = √32 = 4√2.

Using the distance formula: d = √[(6 - 6)² + (2 - 8)²] = √[0 + 36] = √36 = 6.

Standard form of a circle: (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=5.

The directrix of the parabola x^2 = 4py is given by y = -p. Here, p = 3, so the directrix is y = -3.

Since parallel lines have the same slope, the equation is y - 1 = 3(x - 4) which simplifies to y = 3x - 11.

Parallel lines have the same slope. Using point-slope form: y - 2 = 3(x - 1) gives y = 3x - 1.

Since parallel lines have the same slope, the equation is y - 1 = 3(x - 2) which simplifies to y = 3x - 5.

The slope of the perpendicular line is -1/3. Using point-slope form: y - 1 = -1/3(x - 1) gives y = -1/3x + 4/3.

The slope of the perpendicular line is -1/3. Using point-slope form: y - 1 = -1/3(x - 1).

The equation of a line through the origin with slope m is y = mx. Thus, y = -3x.

Using the slope-intercept form y = mx + b, with m = -4 and b = 0, the equation is y = -4x.

In the equation y^2 = 4px, we have 4p = 20, thus p = 5. The focus is at (5, 0).

For the parabola y^2 = 4px, here 4p = 20, so p = 5. The focus is at (5, 0).

The latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 12, so p = 3, and the latus rectum is 4p = 12.

Using the distance formula, length = sqrt((7-3)^2 + (1-4)^2) = sqrt(16 + 9) = 5.

The slope of the line is 4/5, so the slope of the perpendicular line is -5/4.

The slope of the line is 4/5, so the slope of the perpendicular line is -5/4.

Rearranging gives y = 2x + 4, so slope = 2.

In the equation y^2 = 4px, we have 4p = 20, thus p = 5.