Maximum Distance Within a Circle

Mathsnut

New member
Joined
Feb 11, 2020
Messages
2
If lines L1 & L2 form a right angle within a circle, at what point P are the lengths of L1 + L2 maximized?

Note: the right angle is not bound to the circumference, but may be placed at any location within the circle.
 

HallsofIvy

Elite Member
Joined
Jan 27, 2012
Messages
5,641
What points are you taking as the "starting" and "ending" points of the line segments? If the lines start and end on the circle then obviously the lines will be longest if they are diameters of the circle so P is the center of the circle.
 

Cubist

Junior Member
Joined
Oct 29, 2019
Messages
224
graphic.png

HallsofIvy is asking if P=A, L1=ED, L2=CB

My guess is that P=A, L1=AD, and L2=AB.

@Mathsnut please clarify
 

Mathsnut

New member
Joined
Feb 11, 2020
Messages
2
Yes Cubist. L1=AD & L2=AB.
 

lev888

Full Member
Joined
Jan 16, 2018
Messages
913
If lines L1 & L2 form a right angle within a circle, at what point P are the lengths of L1 + L2 maximized?

Note: the right angle is not bound to the circumference, but may be placed at any location within the circle.
I think the idea is to find not the point, but how the lines are oriented relative to the center of the circle.
Pick a point P on the circumference. Draw the 2 lines. Start turning the lines around P. The angle of rotation is your variable. Figure out the length as a function of the angle. Find the maximum.
 

Cubist

Junior Member
Joined
Oct 29, 2019
Messages
224
It does seem a very good bet that P lies on the circumference. If you want to try lev888's suggestion, then I have a hint. There is a very simple polar coordinate equation that gives the following graph:-

20200211_circle3.png

For −π/2 ≤ θ ≤ π/2 and r = ? < a trig function, do you know it? >

Can you see how this helps?

--

However I am also wondering if you are studying partial derivatives in class? If so, then your teacher probably wants you to solve this question in a different way (by considering ALL points within the circle).
 
Top