According to the converse of alternate segment theorem "If a line is drawn through an end point of a chord of a circle so that the angle formed with the chord is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle"

How many tangents can be drawn on the circle from a point outside the circle?

Suppose $S_1 \mathrm{a}\mathrm{n}\mathrm{d} S_2$ are two unequal circles; $AB$ and $CD$ are the direct common tangents to these circles. A transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$ units, then $RS$ is -

On the circle with center $O$, points $A,B$ are such that $OA = AB$. A point $C$ is located on the tangent at $B$ to the circle such that $A$ and $C$ are on the opposite sides of the line $OB$ and $AB=BC$. The line segment $AC$ intersects the circle again at $F$. Then the ratio $\angle BOF:\angle BOC$ is equal to -

At one end $\mathrm{A}$ of a diameter $\mathrm{AB}$ of a circle of radius $5 \mathrm{c}\mathrm{m}$, tangent $\mathrm{XAY}$ is drawn to the circle. The length of the chord $\mathrm{CD}$ parallel to $\mathrm{XY}$ and at a distance $8 \mathrm{c}\mathrm{m}$ from $\mathrm{A}$ is

In the figure below, a circle with centre $\mathrm{O}$ is inscribed in a quadrilateral $\mathrm{ABCD}$ such that, it touches sides $\mathrm{BC}$, $\mathrm{AB}$, $\mathrm{AD}$ and $\mathrm{CD}$ at points $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ and $\mathrm{S}$ respectively. If $\mathrm{A}\mathrm{B}=29 \mathrm{c}\mathrm{m}$, $\mathrm{A}\mathrm{D}=23 \mathrm{c}\mathrm{m}, \mathrm{\angle }\mathrm{B}=90°$ and $\mathrm{D}\mathrm{S}=5 \mathrm{c}\mathrm{m},$ then the radius of the circle (in $\mathrm{cm}$) is

Two circles touch each other externally at $\mathrm{P}$. $\mathrm{AB}$ is a common tangent to the circle touching them at $\mathrm{A}$ and $\mathrm{B}$. Then value of $\mathrm{}\mathrm{\angle }\mathrm{A}\mathrm{P}\mathrm{B}$ is

In a right triangle $\mathrm{ABC}$, right-angled at $\mathrm{B}, \mathrm{B}\mathrm{C}=12 \mathrm{c}\mathrm{m}$ and $\mathrm{A}\mathrm{B}=5 \mathrm{c}\mathrm{m}.$ The radius of the circle inscribed in the triangle (in $\mathrm{cm}$) is

In the figure below, $\mathrm{DE}$ and $\mathrm{DF}$ are tangents from an external point $\mathrm{D}$ to a circle with centre $\mathrm{A}$. If $\mathrm{D}\mathrm{E}=5 \mathrm{c}\mathrm{m}$ and $\mathrm{D}\mathrm{E}\perp \mathrm{D}\mathrm{F},$ then the radius of the circle is

In Fig., $\mathrm{QR}$ is a common tangent to the given circles touching externally at the point $\mathrm{T}$. The tangent at $\mathrm{T}$ meets $\mathrm{QR}$ at $\mathrm{P}$. If $\mathrm{P}\mathrm{T}=3.8 \mathrm{c}\mathrm{m},$ then the length of $\mathrm{QR}$  (in $\mathrm{cm}$) is

​​​​​​​

In the figure, $\mathrm{PQ}$ and $\mathrm{PR}$ are two tangents to a circle with centre $\mathrm{O}$. If $\mathrm{\angle }\mathrm{Q}\mathrm{P}\mathrm{R}=46°,$ then $\mathrm{\angle }\mathrm{Q}\mathrm{O}\mathrm{R}$ is
 

​​​​​​​