Math Is Fun Forum

Identity

Member

Registered: 2007-04-18

Posts: 934

position vector of centre of circle

Relative to an origin, points, A, B, C, D, and E have position vectors:

Find the position vector of the centre of the circle through E, D, B and F (

).

Sorry for not having a diagram, hopefully you can work it out using the vectors themselves. Thanks!

This is actually the last part of a question so here are the other parts in case they might help:
a) Show that E lies on lines DA and BC.
b) Find

and

c) Find the position vector of point F, the intersection of lines DC and AB.
d) Show that FD is perpendicular to EA and EB is perpendicular to AF.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: position vector of centre of circle

Note that the y-axis is the perpendicular bisector of the chord BD. Hence the centre is on the y-axis.

Let the centre be

.

Solve for k.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: position vector of centre of circle

Thanks Jane!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: position vector of centre of circle

You’re welcome.

On second thoughts, however, your having found the point

makes the problem actually much easier. Now, we know that the y-axis is a line of symmetry. Hence the point

, which is the reflection of

in the y-axis, lies on the circle. The perpendicular bisector of the chord E′F is easily seen to be the line

. Hence the centre is the intersection of the y-axis

and the line

.