In the given figure, AB is chord of the circle with centre O, BT is tangent to the circle. The values of x and y are

Given AB is a circle and BT is a tangent,

∠BAO = 32°

Here, ∠ OBT = 90°

[Since tangent is ⊥ to the radius at the point of contact]

OA = OB [ Radii of the same circle ]

∴ ∠ OBA = ∠OAB = 32° [ Angles opposite to equal side are equal]

∴ ∠ OBT = ∠ OBA + ∠ ABT = 90°

⇒ 32° + x = 90°.

⇒ ∠ x = 90° – 32° = 58° .

Also, ∠ AOB = 180° – ∠OAB – ∠OBA

= 180° – 32° – 32° = 116°

Now Y = ½ AOB [ Angle formed at the center of a circle is double the angle formed in the remaining part of the circle] = ½ × 116° = 58° .