Circles

A tangent $PQ$ is drawn from point $P$ to a circle with center $O$ and radius $20\text{ cm}$. If $OP = 29\text{ cm}$, find $PQ$.

If $PA = 3x + 5$ and $PB = 2x + 9$ are lengths of tangents from point $P$ to a circle, then $x$ is:

Two circles of radii $10\text{ cm}$ and $8\text{ cm}$ are concentric. The length of a chord of the larger circle that is tangent to the smaller circle is:

From a point located on the boundary of a circular field, how many straight lines can be drawn that touch the field at exactly one point?

A point $U$ is $5$ cm away from the center of a circle of radius $11$ cm. The number of tangents from $U$ is:

A point $V$ is $15$ cm away from the center of a circle of radius $11$ cm. The number of tangents from $V$ is:

If the radius of a circle is $5 \text{ cm}$ and the distance of a point $T$ from the center is $4 \text{ cm}$, then the number of tangents that can be drawn from $T$ to the circle is:

The angle between two tangents from an external point is ______ to the angle subtended by the line segments joining the points of contact at the center.

The point where a tangent meets the circle is called the point of contact. This point always lies: