Two circles with centres O and O′ intersect at two points A and B. A line PQ is drawn parallel to OO′ through A(or B) intersecting the circles at P and Q. Prove that PQ = 2 OO′.

Answers (1)

Two circles with centres O and O′ intersect at two points A and B.

PQ || OO'

To Prove that: PQ = 2OO'

Proof:

Draw OM perpendicular to PB and O'N perpendicular to BQ

From the figure, we have:

OP = OB  (radius of same circle)

O'B = O'Q (radius of the same circle)

In \triangleOPB,

BM = MP        … (i)   (perpendicular from the centre of the circle bisects the chord)

Similarly in \triangleO'BQ,

BN = NQ         … (ii)  (perpendicular from the centre of the circle bisects the chord)

Adding (i) and (ii),

BM + BN = PM + NQ

Adding BM + BN to both the sides.

BM + BN + BM + BN = BM + PM + NQ + BN.

2BM + 2BN = PQ

2(BM + BN) = PQ      … (iii)

OO' = MN (As OO' NM is a rectangle)

OO' = BM + BN         ... (iv)

Using equation (iv) in (iii)

2 OO'= PQ

Hence proved.

Preparation Products

NEET Foundation 2022

NEET Foundation 2022.

₹ 14999/-
Buy Now
Biology Foundation for Class 10

Biology Foundation for Class 10.

₹ 999/- ₹ 499/-
Buy Now
Knockout JEE Main April 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout NEET May 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout JEE Main May 2021

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 22999/- ₹ 9999/-
Buy Now
Boost your Preparation for JEE Main with our Foundation Course
 
Exams
Articles
Questions