Concepts: the parabola

3.1 Definition and standard equation

A parabola is the set of points in a plane that are at the same distance from a fixed point, the focus, and a fixed line, the directrix. This equal-distance definition produces the curve when the squared distance to the focus is set equal to the squared distance to the directrix, which cancels the square terms and leaves a linear term.

In the standard form y squared equal to 4ax the curve opens towards positive x. Its vertex sits at the origin, the focus is the point (a, 0) and the directrix is the vertical line x equal to minus a. A point on this parabola at parameter t is (a t squared, 2at), and the chord through the focus cuts the parabola at parameter values whose product is minus 1. That property is the reason many focal chord problems reduce to two lines of algebra.

The other orientations are x squared equal to 4ay opening upward, y squared equal to minus 4ax opening to the left and x squared equal to minus 4ay opening downward. In each case the latus rectum is the line segment through the focus perpendicular to the axis with length 4a.

The parametric form also shortens chord and normal questions. For y squared equal to 4ax, the chord joining parameters t1 and t2 has a midpoint whose coordinates are a(t1 squared plus t2 squared) by 2 and a(t1 plus t2). When the chord passes through the focus, the product t1 t2 equals minus 1, so the midpoint lies on a known line. This is the algebra behind most focal chord questions, and it saves solving a quadratic for every intersection point.

3.2 Latus rectum and shifted forms

The latus rectum is the focal chord perpendicular to the axis, and its length for the standard parabola is 4a. Its endpoints are (a, 2a) and (a, minus 2a) for y squared equal to 4ax. A question that gives the latus rectum gives a directly, and a question that gives the focus gives a, so most standard parabola questions start from one of these two readings.

A parabola with vertex at (h, k) and opening to the right has equation (y minus k) squared equal to 4a(x minus h). The focus moves to (h plus a, k) and the directrix moves to x equal to h minus a. Shifting the equation is much easier than expanding it, so keep the squared bracket form in answers.

An important application is the tangency condition. A line y equal to mx plus c touches the parabola y squared equal to 4ax exactly when c equals a by m. For the general orientation the condition changes, so derive it from the discriminant rather than remembering every variant.

The parametric form also shortens chord and normal questions. For y squared equal to 4ax, the chord joining parameters t1 and t2 has a midpoint whose coordinates are a(t1 squared plus t2 squared) by 2 and a(t1 plus t2). When the chord passes through the focus, the product t1 t2 equals minus 1, so the midpoint lies on a known line. This is the algebra behind most focal chord questions in the exam, and it saves solving the quadratic for the intersection points.

The latus rectum also measures how wide the parabola is at the focus. A short latus rectum means a narrow curve and a larger a gives a wider one, which is a quick way to sanity check a computed value against a sketch.

3.3 Exercise 10.2 practice

The opening exercise item works with y squared equal to 12x and wants its focus, axis, directrix and latus rectum. Comparing with y squared equal to 4ax gives a equal to 3, so the focus is (3, 0), the axis is the x-axis, the directrix is x equal to minus 3 and the latus rectum has length 12.

Another item fixes the vertex at the origin and the axis along the x-axis, and sends the curve through the point (2, minus 3). Since the point has a y value whose square is positive and x positive, the parabola opens right and has the form y squared equal to 4ax. Substituting gives 9 equal to 8a, so a is 9 by 8 and the equation is y squared equal to 9x by 2.

Question 7 asks for the equation whose focus is (0, minus 3) and directrix is y equal to 3. The vertex is exactly between them at the origin and the parabola opens downward, so x squared equal to minus 12y, because the distance from vertex to focus is 3 and 4a is 12 with the negative sign for downward opening.