Question

the centres of two circles are P and Q ;they intersect at the points A and B. The straight line parallel to the linesegment PQ through the point A intersects the two circles at the points C and D. Prove that CD =2PQ​

Answer 1

Answer :

Given, CD is parallel to PQ, Two Circles with centers P and Q intersect at point A and B.

To Prove: 2PQ = CD                                                                                                                

Construction: PE and QE are the Perpendiculars on the Chord CD From centers P and Q respectively.

⇒ DE = EA

⇒ CF = AF

⇒ DE + EA + AF + FC = DC

⇒ 2EA + 2AF = DC

⇒ 2(EA + AF) = DC ……..(1)

PQ and CD are parallel lines and PE and QE are the perpendiculars.

So, AEP = 900,AFQ = 900,EPQ = 900,AFQ = 900

EPQF is a rectangle

So, EF = PQ ……… (2)

⇒ 2(EA + AF) = DC

⇒ 2EF = DC

⇒ 2PQ = DC