Equal circles with centres $\mathrm{O}$ and $\mathrm{O}\text{'}$ touch each other at $\mathrm{X}$. $\mathrm{OO}\text{'}$ produced to meet a circle with centre $\mathrm{O}\text{'}$, at $\mathrm{A}$. $\mathrm{AC}$ is a tangent to the circle whose centre is $\mathrm{O}$. $\mathrm{O}\text{'}\mathrm{D}$ is perpendicular to $\mathrm{AC}$. Find the value of $\frac{\mathrm{D}\mathrm{O}\mathrm{\text{'}}}{\mathrm{C}\mathrm{O}}.$

In the figure below, $\mathrm{BC}$ is a tangent to the circle with centre $\mathrm{O}$. $\mathrm{OE}$ bisects $\mathrm{AP}$. Prove that $\mathrm{\Delta }\mathrm{A}\mathrm{E}\mathrm{O}\mathrm{}~\mathrm{\Delta }\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}.$

In Fig., $\mathrm{P}\mathrm{O}\perp \mathrm{Q}\mathrm{O}.$ The tangents to the circle at $\mathrm{P}$ and $\mathrm{Q}$ intersect at a point $\mathrm{T}$. Prove that $\mathrm{PQ}$ and $\mathrm{OT}$ are right bisector of each other.

In the given figure, $\mathrm{O}$ is the centre of the circle and $\mathrm{BCD}$ is tangent to it at $\mathrm{C}$. Prove that $\mathrm{\angle }\mathrm{B}\mathrm{A}\mathrm{C}+\mathrm{\angle }\mathrm{A}\mathrm{C}\mathrm{D}=90°$.

In the figure, there are two concentric circles with centre $\mathrm{O}$. $\mathrm{PRT}$ and $\mathrm{PQS}$ are tangents to the inner circle from a point $\mathrm{P}$ lying on the outer circle. If $\mathrm{P}\mathrm{R}=5 \mathrm{c}\mathrm{m},$ find the length of $\mathrm{PS}$.

In the given figure, $\mathrm{PQ}$ is a tangent from an external point $\mathrm{P}$ to a circle with centre $\mathrm{O}$ and $\mathrm{OP}$ cuts the circle at $\mathrm{T}$ and $\mathrm{QOR}$ is a diameter. If $\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{R}=130°$ and $\mathrm{S}$ is a point on the circle, find $\angle 1+\angle 2$ .

In the figure below, $\mathrm{PA}$ and $\mathrm{PB}$ are tangents to the circle from an external point $\mathrm{P}$. $\mathrm{CD}$ is another tangent touching the circle at $\mathrm{Q}$. If $\mathrm{P}\mathrm{A}=12 \mathrm{c}\mathrm{m}, \mathrm{Q}\mathrm{C}=\mathrm{Q}\mathrm{D}=3 \mathrm{c}\mathrm{m},$ then find $\mathrm{P}\mathrm{C}+\mathrm{P}\mathrm{D}$.

In the figure below, $\mathrm{PA}$ and $\mathrm{PB}$ are tangents to the circle drawn from an external point $\mathrm{P}$. $\mathrm{CD}$ is a third tangent touching the circle at $\mathrm{Q}$. If $\mathrm{P}\mathrm{B}=10 \mathrm{c}\mathrm{m}$ and $\mathrm{C}\mathrm{Q}=2 \mathrm{c}\mathrm{m},$ what is the length $\mathrm{PC}$ ?