Question

If a line is drawn from the center of a circle to the midpoint of a chord, prove by vector method that the line is perpendicular to the chord.

Answer 1

\[OM + MA =OA\] (1)
\[OM + MB = OB\]
\[OA - MA = OB\] (2)
Squarring (1) and (2) scalarly and subtracting, we have
\[4OM.MA=0,[OA.OA=OA ^{2},etc]\]
OM is perpendicular to MA i.e., OM is perpendicular to MA
or OM is perpendicular to AB..

This is the solution given in my textbook but I can't understand the equation (2)..

Answer 2

I have ignored the arrows in the vectors..

Answer 3

AP=PB = a/2
By triangle law,
r1=p-(a/2)...1
r2=p+a/2..2(all vectors)
now take modulus of both sides and square
\[r^2=p^2+a^2/4-pacostheta\]...1
r^2=p^2+a^4+pacostheta...
i used \[\left| a+b \right|^2=a^2+b^2+2ab \cos \theta\]
now add 1 and 2
r^2=p^2+a^2/4
pythagorean theorem