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

The equation x^2 = 12y can be rewritten as (y - 0) = (1/3)(x - 0)^2, indicating the focus is at (0, 3).

The standard form of a parabola is y^2 = 4px. Here, 4p = 12, so p = 3. The focus is at (p, 0) = (3, 0).

The length of the latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 16, so p = 4. Therefore, the length of the latus rectum is 4 * 4 = 16.

Length = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.

Length = √[(4-1)² + (5-1)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.

Length = |5 - 2| = 3.

Midpoint = ((1+3)/2, (2+4)/2) = (2, 3).

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 4)/2, (3 + 7)/2) = (3, 5).

Setting 2x + 1 = -x + 4 gives 3x = 3, thus x = 1. Substituting x back gives y = 3, so the point is (1, 3).

Set x + 1 = -x + 5. Solving gives x = 2, y = 3. Thus, the point is (2, 3).

The slope m is given by (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2.

Rearranging gives y = (2/3)x + 2, so slope = 2/3.

The slope m = (5 - 0) / (5 - 0) = 1.

The slopes can be found by solving the quadratic equation for m in terms of x and y.

The slopes can be calculated from the quadratic equation, yielding slopes of 5/6 and -1.

Set x = 0: -2y = 10 => y = -5. The y-intercept is (0, -5).

The semi-major axis is 4 and the semi-minor axis is 3 after rewriting the equation in standard form.

The distance between the foci of the hyperbola is 2c, where c = √(a^2 + b^2) = √(25 + 16) = √41, so the distance is 2√41.

The slopes can be found by solving the quadratic equation derived from the given equation.

The product of the slopes of the lines can be found from the equation, which gives -1.

The sum of the slopes can be found using the relationship between the coefficients of the quadratic.

The condition for parallel lines is that the determinant of the coefficients must equal zero.

Factoring the equation gives the slopes as m1 = 1 and m2 = 3.

The lines are at an angle of 45 degrees as the determinant of the coefficients gives a non-zero value.

The sum of the slopes is given by - (coefficient of xy)/(coefficient of x^2) = -6/5.

The sum of the slopes of the lines is given by -b/a, which is -5/6.

The slopes can be found by solving the quadratic equation formed by the coefficients of x^2, xy, and y^2.

The angle can be calculated using the slopes derived from the equation.

The vertex of the parabola y^2 = 4px is at (0, 0).