Question

In the given figure, $XY$ and $X^1$$Y^1$ are two parallel tangents to a circle with centre O and another tangent $AB$ with point of contact C intersecting $XY$ at $A$ and $X^1$$Y^1$ at $B$ then $\angle AOB$ is ___.

• A. ${100}^{\circ }$
• B. ${60}^{\circ }$
• C. ${80}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is D ${90}^{\circ }$


In the figure, if we join point 'O' and 'C'. Then we can see that OC = DA = EB,
OE = CB and OD = CA
$\because$ OC = OD = OE [radius of circle]
$\therefore$ ODAC and OEBC are squares

$\because$ By Theorem - The centre of a circle lies on the bisector of the angle between two tangents drawn from a point outside it.

$\therefore$ AO bisects $\angle$CAD and BO bisects $\angle$CBE.
This means AO and BO are diagonals of square ODAC and OBEC respectively.
So, $\angle$AOC = ${45}^{\circ }$ and $\angle$BOC = ${45}^{\circ }$.

Now, in $△$ AOB
$\angle$ AOB = $\angle$AOC + $\angle$BOC
$\Rightarrow$ $\angle$ AOB = ${45}^{\circ }$ + ${45}^{\circ }$ = ${90}^{\circ }$