The length of the latus rectum for the parabola x^2 = 4py is given by 4p. Here, 4p = 16, so p = 4. Thus, the length of the latus rectum is 4p = 16.

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

Rewriting gives x^2/9 + y^2/4 = 1. Here, a^2 = 9, b^2 = 4, c = √(a^2 - b^2) = √(9 - 4) = √5. Eccentricity e = c/a = √5/3 ≈ 0.6.

The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is y^2/a^2 + x^2/b^2 = 1.

A represents the amplitude of the oscillation, which is the maximum displacement from the mean position.

In the equation of motion for simple harmonic motion, φ is the phase constant, which determines the initial position of the oscillator.

The equation of state for an ideal gas is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.

The directrix of the parabola y^2 = 8x is given by x = -2.

Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.

Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.

For two lines to be perpendicular, the product of their slopes must equal -1.

The correct form of the equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.

The equation of the lines can be expressed as y = (m1 + m2)x, representing the sum of the slopes.

The slope of the tangent at x = 2 is f'(x) = 2x, so f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 4x - 8.

The derivative f'(x) = 2x. At x = 2, f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 2x - 4.

This is a perfect square: (x + 1)^2 = 0.

The discriminant is 0, indicating one repeated root.

The discriminant is 0, indicating one repeated root.

For both roots to be positive, k must be greater than the square of half the coefficient of x: k > (6/2)^2 = 9.

Factoring the equation gives (x - 2)(x - 5) = 0, so the roots are 2 and 5.

Factoring gives (x-3)(x+3)=0, hence the roots are -3 and 3.

Kp can be calculated from Kc using the relation Kp = Kc(RT)^(Δn), where Δn is the change in moles of gas. Here, Δn = 2 - 3 = -1, so Kp = Kc(0.0821T)^(-1). Assuming standard conditions, Kp = 4 * (1/RT) = 16.

The escape velocity from the surface of the Earth is approximately 11.2 km/s.

The establishment of Yellowstone National Park in 1872 marked the beginning of the national park movement in the United States, influencing conservation efforts worldwide.

The Aligarh Movement, led by Sir Syed Ahmed Khan, aimed to promote modern scientific education among Muslims in India.

4^(x+1) = (2^2)^(x+1) = 2^(2(x+1)) = 2^(2x+2).

The equation x^2 + y^2 = r^2 represents a circle with radius r.

The equation y = a(x - h)^2 + k represents a quadratic function in vertex form.

The equation y = a(x - h)^2 + k represents a family of parabolas with vertex (h, k).

The equation y = ax^2 + bx + c represents a quadratic function.