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

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

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.

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

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

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

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

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

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

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

Rearranging to slope-intercept form: -2y = -5x + 10, thus y = (5/2)x - 5. The y-intercept is -5.

Substituting x = 1 into the equation gives y = 1^2 - 4 = -3, so the point (1, -3) lies on the parabola.