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Q.

State true or false.


In the given figure, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Here, ∠ AOB = 90°.


In the given figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C, is intersecting XY at A and

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a

True

b

False 

answer is A.

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Detailed Solution


Given that XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B.
In the given figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C, is intersecting XY at A and Join OC.
OC is perpendicular to AB (radius of the circle is perpendicular to its tangent)
Therefore,
∠ACO=∠BCO= 90 °  
In ΔAOP and ΔAOC  
AO = AO (Common side)
OP=OC   (radius)
AP=AC   (the length of tangents drawn from external point are equal)
Therefore, ΔAOP≅ΔAOC  
⇒∠AOP=∠AOC  … (1)
In ΔBOC and ΔBOQ  
OB = OB  (common side)
OC=OQ   (radius)
BC=BQ   (Theorem 2)
Therefore, ΔBOC≅ΔBOQ  
⇒∠BOC=∠BOQ   .… (2)
For line PQ, ∠AOP+∠AOC+∠BOC+∠BOQ= 180 °  
Substitute the equations (1) and (2) in the above equation,
we get,
⇒∠AOC+∠AOC+∠BOC+∠BOC= 180 ° ⇒2∠AOC+2∠BOC= 180 ° ⇒2(∠AOC+∠BOC)= 180 °  
⇒∠AOC+∠BOC= 180 ° 2 ⇒∠AOB= 90 °  
Hence, ∠AOB= 90 °  
Therefore, the given statement is true.
Hence, option (1) is correct.
 
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