Focal Diameter Of A Parabola

A parabola is a locus of points equidistant from both one) a single point, called the focus of the parabola, and 2) a line, called the directrix of the parabola.

What is the Focus and Directrix?

The scarlet point in the pictures below is the focus of the parabola and the ruby-red line is the directrix. Equally you can see from the diagrams, when the focus is above the directrix Example 1, the parabola opens upwards. In the next department, we will explicate how the focus and directrix relate to the actual parabola. Explore this more with our interactive app below.

Instance 1 Picture of the focus and Directrix of a parabola Example 2

When the focus is below, the directrix , so the parabola opens downwards.

picture of focus and directrix negative a

How do Focus/Directrix relate to the Parabola?

The purple lines in the picture beneath correspond the distance between the focus and unlike points on the directrix . Every bespeak on the parabola is simply as far away (equidistant) from the directrix and the focus.

Picture of focus and directrix as locus

In other words, line $$ l_1 $$ from the directrix to the parabola is the aforementioned length as $$ l_1 $$ from the parabola back to the focus . The same goes for all of the other distances from a point on the parabola to the focus and directrix ( $$ l_2, l_3 \text{ etc.. } $$). Run across animation below

Explore this more with our interactive app below.

Parabola Locus Animation

Exploring Focus/Directrix relation to Graph

You probably know that the smaller |a| in the standard class equation of a parabola, the wider the parabola. In other words y = .1x² is a wider parabola than y = .2x² and y = -.1x² is a wider parabola than y = .-2x². Y'all can understand this 'widening' effect in terms of the focus and directrix. As the distance betwixt the focus and directrix increases, |a| decreases which ways the parabola widens. Come across the pictures beneath to understand.

|a| = ane

|a| = .6

|a| =.iii

|a| = .2

Focus and Directrix Applet

Explore how the focus and directrix relate to the graph of a parabola with the interactive programme below.

oy2%3Asvi1y2%3Artzy2%3Afdi1y2%3Alci1y2%3Aaxi1y2%3Ayizy2%3Asgzy2%3Agazy3%3Apoai1y3%3Apobi2y3%3Apoci-3y3%3Ashxi51y3%3Ashyi-176y1%3Azi2g

Try this interactive parabola applet on its own page.