Question

In the given figure, O is the centre of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circles, respectively. If PA = 10 cm, find the length of PB up to one decimal place.

Answer 1

$$\begin{gathered} \text{Given,}O\text{is the centre of two concentric circles of radii}OA=6\text{cm and}OB=4\text{cm}\text{.} \\ PA\text{and}PB\text{are the two tangents to the outer and inner circles respectively and}PA=10\text{cm}\text{.} \\ \text{Now, tangent drawn from an external point is perpendicular to the radius} \\ \text{at the point of contact}\text{.} \\ \therefore \angle OAP=\angle OBP={90}^0 \\ \therefore \text{From right-angled}△OAP,O{P}^2=O{A}^2+P{A}^2 \\ \begin{array}{l}=>OP=\sqrt{O{A}^2+P{A}^2}\\ =>OP=\sqrt{6^2+{10}^2}\\ =>OP=\sqrt{136}\text{cm}\text{.}\end{array} \\ \therefore \text{From right-angled}△OAP,O{P}^2=O{B}^2+P{B}^2 \\ \begin{array}{l}=>PB=\sqrt{O{P}^2-O{B}^2}\\ =>PB=\sqrt{136-16}\\ =>PB=\sqrt{120}\text{cm}\\ \text{=>}PB=10.9\text{cm}\text{.}\end{array} \\ \therefore \mathrm{The}\mathrm{length}\mathrm{of}PB\mathrm{is}10.9\mathrm{cm}. \end{gathered}$$