Question

Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord

Answer 1

A circle with centre at O.

AB is chord of the circle and OP bisects AB

(ie) AP = PB

To prove `bar"OP"` ⊥ `bar"AB"` O is the position vector

∴ `bar"OA" = bar"OB"` = Radius

Position vector of P

`bar"OP" = (bar"OA" + bar"OB")/2`

`bar"OP"*bar"AB" = bar"OP" * (bar"OB" - bar"OA")`

= `((bar"OB" + bar"OA")/2)* (bar"OB" - bar"OA")`

= `1/2[|bar"OB"|^2 - |bar"OA"|^2]`

= 0    ........`(∵ bar"OA" = bar"OB" = "Radius")`

∴ `bar"OP"` ⊥ `bar"AB"`

Hence proved