A parabolic curve is often used in the design of racing tracks - Leaving Cert DCG - Question A-3 - 2010

The latus rectum of the parabola is a line segment through the focus, perpendicular to the axis of symmetry. It can be calculated based on the equation of the parabola. For points clearly outside the latus rectum, consider points like (2, 4) which, being away from the path of the parabola, can be found by plugging in the coordinates into the parabola equation derived from the standard form.

For locating a point inside the latus rectum, we identify coordinates that are closer to the vertex (0, 0) but within the curve path. For example, points like (1, 1) will be located inside as they are below the latus rectum boundary.

The equation of the parabola can be expressed as:

$y = \frac{1}{4p}x^2$

Assuming p = 1, the relation becomes:

$y = \frac{1}{4}x^2$

Sketch the curve using this equation to show the parabolic shape extending through points such as the vertex and focus.

To find a point that is 45mm (or 4.5 units) away from the focus F(0, 1), we consider both directions vertically. We can find the points as:

1. Directly above: (P_1 = (0, 1 + 4.5) = (0, 5.5))
2. Directly below: (P_2 = (0, 1 - 4.5) = (0, -3.5))

Both points satisfy the distance from the focus.

The tangent to the parabola at any point can be derived from the parabola's equation by using the derivative to find the slope at that specific point. For the point (3, 1), calculating the slope will yield the tangent line equation. Thus, the equation of the tangent can be constructed:

1. Find the derivative of the parabola at (3, 1).
2. Use the point-slope form to derive: $y - 1 = m(x - 3).$