Question : In the given figure, if $\mathrm{PA}$ and $\mathrm{PB}$ are tangents to the circle with centre $\mathrm{O}$ such that $\angle \mathrm{APB}=54^{\circ}$, then $\angle \mathrm{OBA}=$________.
Option 1: 27°
Option 2: 40°
Option 3: 30°
Option 4: 35°
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Correct Answer: 27°
Solution :
We have to find $\angle AOB$.
In quadrilateral APBO,
$\angle A = \angle B = 90°$
Now, $\angle P + \angle A + \angle B + \angle O = 360°$
⇒ $54° + 90° + 90° + \angle O = 360°$
⇒ $\angle O = 126°$
Now, in $\triangle AOB$
$\angle OBA + \angle OAB + \angle AOB = 180°$
Now, $\angle OBA$ and $\angle OAB$ will be equal as the tangents subtends equal angles.
⇒ $2\angle OBA = 180° - 126°$
⇒ $2\angle OBA = 54°$
⇒ $\angle OBA = 27°$
Hence, the correct answer is 27°.
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