In the adjoining figure, XY and X'Y' are parallel tangents to a circle with centre O . Another tangent AB touches the…

CBSE Class 10 Maths PYQ · Circles · Triangle & Circle · 3 Marks · March 2025 · Standard

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993 Marks · March 2025 · Standard

In the adjoining figure, $XY$ and $X'Y'$ are parallel tangents to a circle with centre $O$. Another tangent $AB$ touches the circle at $C$ intersecting $XY$ at $A$ and $X'Y'$ at $B$. Prove that $AB$ subtends right angle at the centre of the circle; or $\angle AOB = 90^\circ$.

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Join $OC$.
$\Delta POA \cong \Delta COA$
$\angle POA = \angle COA$
Similarly, $\angle QOB = \angle COB$
$\angle POA + \angle QOB + \angle COA + \angle COB = 180^\circ$
$\implies 2(\angle COA + \angle COB) = 180^\circ$
$\implies \angle COA + \angle COB = 90^\circ$
$\therefore \angle AOB = 90^\circ$