AB and CD are two chords of a circle which intersect at a point O inside the circle. It is given that, AB = 10 cm, CO = 1.5 cm and DO = 12.5 cm. What is the ratio between the larger and smaller among AO and BO?
Show answer & explanation
Correct answer: C
Explanation
Correct Answer is option 3 i.e. 3 : 1
Concept:
If 2 chords AB and CD of a circle intersect each other at O:
AO × OB = CO × OD
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Understanding |
Application |
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Given: OC = 1.5, OD = 12.5, AB = 10 Suppose AO = x Then, OB = (10 – x)
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AO × OB = CO × OD ⇒ x × (10 – x) = 1.5 × 12.5 ⇒ 10x – x2 = 18.75 ⇒ x2 – 10x + 18.75 = 0 Multiplying by 4: ⇒ 4x2 – 40x + 75 = 0 ⇒ 4x2 – 30x – 10x + 75 = 0 ⇒ 2x (2x – 15) – 5(2x – 15) = 0 ⇒ (2x – 5) (2x – 15) = 0 ⇒ x = 5/2 or x = 15/2 When x = 5/2: (10 – x) = (10 – 5/2) = 15/2 When x = 15/2: (10 – x) = (10 – 15/2) = 5/2 Hence, The ratio of the larger and smaller among AO and BO = 15/2 : 5/2 = 3 : 1 |
