How to construct a parabola given its focus and two points on it

Question

I need to construct the guideline/directrix and the axis of symmetry of a parabola.

I’ve been given A its focus and two points B and C going through the parabola.

Could you please point me to the technique on how to do this? Thanks a lot!

Intelligenti pauca · Accepted Answer · 2019-11-08T18:12:24.487

3

Draw two circles, having $B$ and $C$ as centers, and passing through $A$. The directrix of the required parabola is any common tangent to both circles.

edited Nov 08 '19 at 18:12

answered Nov 08 '19 at 17:46

Intelligenti pauca

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What about the axis of symmetry? – Narasimham Nov 08 '19 at 18:35
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@Narasimham Just drop the perpendicular from focus to directrix. – Intelligenti pauca Nov 08 '19 at 18:43
I would draw them both in either case of solution construction, – Narasimham Nov 09 '19 at 02:13
Included the parabola axes in your construction. It is not any new construction, just that I felt the axis should be also added as OP wanted.The method follows the way eccentricity =1 is defined as locus for parabola between focus/directrix. – Narasimham Nov 09 '19 at 19:06
Thank you! Very helpful – ralphjsmit Nov 09 '19 at 19:41

Narasimham · Answer 2 · 2019-11-09T19:10:21.357

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Repeat of Aretino's construction, including axes of symmetry.Points $(p,q)$ are vertices of the two parabolas possible with two direct tangent directrices shown.

edited Nov 09 '19 at 19:10

answered Nov 09 '19 at 18:58

Narasimham

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Thank you, also for correcting the question! – ralphjsmit Nov 09 '19 at 19:41

How to construct a parabola given its focus and two points on it

2 Answers2