The angle between the lines can be found using the formula tan(θ) = |(m1 - m2) / (1 + m1*m2)|, where m1 and m2 are the slopes of the lines. The slopes can be found from the equation.

The slopes are m1 = 2 and m2 = -0.5. The angle θ is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)| = |(2 + 0.5) / (1 - 1)|, which is undefined, indicating 90 degrees.

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

For the lines to be parallel, the discriminant D must be equal to 0.

The condition for the lines to be parallel is given by h^2 = ab.

The condition for the lines to be perpendicular is given by the relation ab + h^2 = 0.

Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3) = (9/3, 6/3) = (3, 2).

Centroid = ((1+3+5)/3, (2+4+6)/3) = (3, 4).

The equation y^2 = -12x can be rewritten as (y - 0)^2 = 4p(x - 0) with p = -3, so the focus is at (-3, 0).

Using the formula for foot of perpendicular, we find the coordinates to be (1, 1).

Using the formula for foot of perpendicular, we find the coordinates to be (3, 2).

For the parabola y^2 = 4px, here 4p = -8, so p = -2. The directrix is given by x = -p, which is x = 2.

Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 3.6, closest option is 4.

Distance = √[(5-2)² + (7-3)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.

Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |3(1) + 4(2) - 12| / sqrt(3^2 + 4^2) = |3 + 8 - 12| / 5 = 1.

Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |2*3 + 3*4 - 6| / sqrt(2^2 + 3^2) = |6 + 12 - 6| / sqrt(13) = 12 / sqrt(13).

Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 5².

The equation y = mx + c represents a family of straight lines where m is the slope and c is the y-intercept.

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

The slope of the perpendicular line is -1/5. Using point-slope form: y - 3 = -1/5(x - 2) gives y = -1/5x + 13/5.

The slope of the given line is 5. The slope of the perpendicular line is -1/5. Using y = mx + c, we get y = -1/5x.

Using the slope-intercept form: y = mx + b, where b = 0, we have y = -2x.

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

Using point-slope form: y - 3 = -1(x - 2) => y = -x + 5.

Using the quadratic formula for the slopes gives m1 = -2 and m2 = -1/3.

Factoring the equation gives (x - 2y)(x + 2y) = 0, which represents the lines x = 2y and x = -2y.

Using the vertex form and substituting the point (2, -4), we find that the equation is y = -2x^2.

The distance from the focus to the directrix is 6, so p = -3. The equation is x^2 = 4py, which gives x^2 = -12y.

The vertex is at (0, 0) and p = 2. The equation is y^2 = 4px, which gives y^2 = 8x.

The vertex form of a parabola is given by (x - h)^2 = 4p(y - k). Here, h = 2, k = 3, and p = 1 (distance from vertex to focus). Thus, the equation is (x - 2)^2 = 4(1)(y - 3) or y = (1/4)(x - 2)^2 + 3.