question_answer

In Fig. 8, O is the centre of a circle of radius 5 cm. T is a point such that \[OT=13\text{ }cm\] and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.

Answer:

Given, a circle with centre of radius 5 cm and \[OT=13\text{ }cm\]

Since, PT is a tangent at P and OP is a radius through P

\[\therefore OP\bot PT\]

In \[\Delta \,OPT\]

\[{{(PT)}^{2}}={{(OT)}^{2}}-{{(OP)}^{2}}\]

\[\Rightarrow PT=\sqrt{{{(13)}^{2}}-{{(5)}^{2}}}\]

\[\Rightarrow PT=\sqrt{169-25}=\sqrt{144}\]

\[\Rightarrow PT=12\,\,cm\]

And      \[TE=OT-OE=(13-5)cm=8\,cm\]

Now,                 \[PA=AE\]

Let                    \[PA=AE=x\]

Then, in \[\Delta \,AET\]

\[{{(AT)}^{2}}={{(AE)}^{2}}+{{(ET)}^{2}}\]

\[{{(12-x)}^{2}}={{(x)}^{2}}+{{(8)}^{2}}\]

\[144+{{x}^{2}}-24x={{x}^{2}}+64\]

\[24x=80\]

\[\Rightarrow AE=x=3.33\,cm\]

\[\therefore AB=2AE=2\times 3.33\]

\[=6.66\,\,cm\]