Question

Let AB be a chord of length 12 of the circle
$\begin{array}{l}(x-2)^2 + (y+1)^2=\frac{169}{4}.\end{array}$
If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to _____ .

Answer 1

Here, AM = BM = 6
$\begin{array}{l}OM = \sqrt{\left(\frac{13}{2}\right)^2-6^2}=\frac{5}{2}\end{array}$
$\begin{array}{l}\sin \theta =\frac{12}{13}\end{array}$
In ΔPAO,
$\begin{array}{l}\frac{PO}{OA}=\sec \theta\end{array}$
$\begin{array}{l}PO = \frac{13}{2}\cdot \frac{13}{5}=\frac{169}{10}\end{array}$
$\begin{array}{l}\therefore PM = \frac{169}{10}-\frac{5}{2}=\frac{144}{10}=\frac{72}{5}\end{array}$
$\begin{array}{l}\therefore 5PM = 72\end{array}$