Question

In the given figure, AB is a chord of length $16cm$ of the circle of radius $10cm$. The tangents at A and B intersect at point P. Find the length of PA.

Answer 1

Join $OA$

$OA=10cm$

$OL$ is perpendicular to $AB$

$AL=LB=\frac{16}{2}=8cm$ [since $OL$ bisects $AB$]

By pythagoras theorem, in $△OLA$

${OL}^2+{LA}^2={OA}^2$

$\Rightarrow {OL}^2={OA}^2-{LA}^2$

$\Rightarrow {OL}^2={10}^2-8^2$

$\Rightarrow {OL}^2=100-64=36$

$\therefore OL=6cm$

$\tan \angle AOL=\frac{AL}{OL}=\frac{8}{6}=\frac{4}{3}$

From $△OAP$

$\tan \angle AOP=\frac{PA}{OA}$

$\Rightarrow \frac{4}{3}=\frac{PA}{10}$

$\Rightarrow PA=\frac{40}{3}$cm

$\therefore$ Length of $PA=\frac{40}{3}\approx 13.33$cm