Using the distance formula: d = √[(1 - 1)² + (5 - 2)²] = √[0 + 9] = √9 = 3.

Using the distance formula: d = √[(2 - 6)² + (3 - 8)²] = √[16 + 25] = √41.

The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan⁻¹(|(m1 - m2) / (1 + m1*m2)|) = tan⁻¹(5/3), which is approximately 90 degrees.

Using the distance formula: d = √[(2 - 2)² + (-1 - 3)²] = √[0 + 16] = √16 = 4.

Setting y = 0 in the equation gives 5x = 10, thus x = 2. The x-intercept is 2.

Using the distance formula: d = √[(2 - (-1))² + (2 - (-1))²] = √[9 + 9] = √18 = 3√2.

Using the distance formula: d = √[(4 - (-2))² + (5 - (-3))²] = √[(4 + 2)² + (5 + 3)²] = √[36 + 64] = √100 = 10.

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

The slope m = (7 - 3) / (4 - 2) = 2. Using point-slope form: y - 3 = 2(x - 2) gives y = 2x + 1.

Solving the equations simultaneously, we find the intersection point is (1, 1).

In the equation x^2 = 4py, we have 4p = 16, thus p = 4. The distance from the vertex to the focus is 4.

'c' represents the y-intercept of the line, which is the point where the line crosses the y-axis.

Setting x = 0 in the equation gives y = 8. Thus, the y-intercept is -8.

The y-intercept is the constant term in the equation, which is 6.

The standard form of a parabola that opens to the left is y^2 = -4px.

Substituting x = 2: 2(2) + 3y = 12 => 4 + 3y = 12 => 3y = 8 => y = 8/3 ≈ 2.67, closest is 3.

Substituting x = 0: 3(0) + 4y = 12 => 4y = 12 => y = 3.

The distance from the focus to the directrix is 4, so the equation is x^2 = 8y.

Substituting x = 0: 4(0) - 5y = 20 => -5y = 20 => y = -4.

Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).

Using point-slope form: y - 3 = 2(x - 1) => y = 2x + 1.

The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.

To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Plugging x = 1 into the equation gives y = -1.

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.