The centers of two circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2,$ each of unit radii, are at a distance of $6$ units from each other. Let $\mathrm{P}$ be the mid-point of the line segment joining the centers of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2.$ A third circle $\mathrm{C}$ is drawn such that it is externally tangent to both the circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ at the same points where the lines passing through $\mathrm{P}$ are tangent to the circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2.$ The radius of the circle $\mathrm{C}$ is

In the given diagram $\mathrm{PQ}$ is parallel to $\mathrm{RS}.$ For $\mathrm{PQ}>\mathrm{OP},$ when $\angle \mathrm{POQ}$ and $\angle \mathrm{OPQ}$ have integral values, find the greatest possible value of $\mathrm{OPQ}.$

In the following figure, $\mathrm{BC}$ and $\mathrm{RP}$ are tangents to the same circle with centre $\mathrm{O}$ and its diameter $\mathrm{AB}.$ Points $\mathrm{R}$ and $\mathrm{S}$ are the points on the tangent $\mathrm{BC},$ such that $\mathrm{AS}$ passes through $\mathrm{Q},$ where $\mathrm{RP}$ touches the circle.

Which of the following is necessarily a correct relations?

In the concerned diagrams, chords $\mathrm{AB}$ and $\mathrm{CD}$ are parallel and radius $\mathrm{OR}$ bisects the chord $\mathrm{AB}$ at $\mathrm{P}$ and $\mathrm{CD}$ at $\mathrm{Q}.$ Radius $\mathrm{OC}$ when extended to a point $\mathrm{T}$ in order to meet the tangent $\mathrm{TR}$ it intersects the chord $\mathrm{AB}$ at S. If $\mathrm{OP}: \mathrm{OS}=8: 17,$ find $\mathrm{OC}: \mathrm{OT}$

Consider the following figure in which $\mathrm{OA}=\mathrm{OB}=\mathrm{AB}.$ Chord $\mathrm{AB}$ is the longest possible chord of the smaller circle and $\mathrm{O}$ is the centre of the larger circle. Find the area of the shaded region, if the radius of the smaller circle is $6 \mathrm{cm}.$

In the following figure $∆\mathrm{ABC}$ is an isosceles triangle, where $\mathrm{AB}=\mathrm{AC}$ and $\mathrm{BC}=2\sqrt{2} \mathrm{cm}. \mathrm{CD}$ is perpendicular on the diameter $\mathrm{AB}$ which intersects the circle at $\mathrm{E}.$ Find $\mathrm{BE}.$

Two parallel chords in a circle have lengths $12$ and $24,$ and the distance between them is $18.$ The chord parallel to these chords and midway between them is of length $\sqrt{\mathrm{a}}$ where $\mathrm{a}$ is

$\mathrm{ABCD}$ is a square of area $16 \mathrm{sq}. \mathrm{cm}$ and $\mathrm{AB}$ is the diameter of the semicircle. The line segments $\mathrm{CP}$ and $\mathrm{DQ}$ intersect at $\mathrm{R}$ and touch the semicircle. Find the area of $∆\mathrm{DRC}.$

In the following figure, a circle of radius $40 \mathrm{mm}$ is inscribed in a square $\mathrm{ABCD}$ in which $\mathrm{M}$ is the mid-point of $\mathrm{AB}.$ If the line segment $\mathrm{MC}$ intersects the circle at $\mathrm{P},$ find $\mathrm{PC}.$