Two circles whose centres are O and O' intersect at P Through P, line "l" parallel to OO' intersecting - Maths - Circles

Question

Two circles whose centres are O and O' intersect at P Through P,
line "l" parallel to OO' intersecting the circles at C and D is
drawn Prove that CD=2OO'

Manish Bharti · Accepted Answer

To Prove : CD = 2OO' Construction: 1 Draw perpendicular PG from point P on OO' 2 Draw perpendiculars OE and O'F from O and O' on CD Proof: CE=EP ( perpendicular from the centre of a circle to a chord bisects the chord) Similarly, PF=FD Since, OO' || CD ( given ) and OE || GP || O'F ( all are perpendiculars on the line CD ) $\Rightarrow$ EP = OG and PF = GO' Therefore, $O O' = O G + G O' = E P + P F O O' = \frac{1}{2} C P + \frac{1}{2} P D = \frac{1}{2} (C P + P D) = \frac{1}{2} C D \Rightarrow 2 O O' = C D$

Two circles whose centres are O and O' intersect at P. Through P, line "l" parallel to OO' intersecting the circles at C and D is drawn. Prove that CD=2OO'.