Question

two circles with centres O and O' intersect at two points A and B. a line PQ is drawn parallel to OO' through A intersecting the circles at P and Q. prove that PQ=2OO'

Answer 1

CD = 2OO'  

Step-by-step explanation:

Here CD is taken instead of PQ

Lets draw OA ⟂ CD and O'B  ⟂ CD

OA  ⟂ CD

⇒ CA = AP = 1/2 AP   ( as perpendicular from center bisect the chord)

⇒  CP = 2AP  

Similarly,  O'B  ⟂ CD

⇒ PB = BD = 1/2 PD

⇒ PD = 2PB

CD = CP + PD

= 2AP + 2PB = 2(AP + PB) = 2AB

AB = OO'  

CD = 2OO'  

QED

Proved

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Answer 2

$\Large{\underline{\underline{\bf{Given:-}}}}$

Two circles whose centres are O and O' intersect at P. Through P a line l parallel to OO' interesting the circles at C and D is drawn.

$\Large{\underline{\underline{\bf{To\: Prove:-}}}}$

CD = 2OO'

$\Large{\underline{\underline{\bf{Construction:-}}}}$

Draw OA ⊥ l and OB ⊥ l

$\Large{\underline{\underline{\bf{Solution:-}}}}$

⇒OA ⊥ l

⇒OA ⊥ CP

⇒CA = AP

⇒CP = 2AP................(1)

and , O'B ⊥ l

⇒O'B ⊥ DP

⇒BP = BD

⇒PD = 2PB..................(2)

∴ CD = CP + PD

By using (I) and (2)

⇒CD = 2AP + 2PB

⇒CD = 2(AP + PB) = 2AB = 2OO'

[∵ ABO'O is a rectangle]

Hence proved!

$\Large{\underline{\underline{\bf{Thanks}}}}$