Question

Two circles of radii $5cm$ and $3cm$ intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

Answer 1

$\begin{array}{c}QL=xthen{O}^{\prime }L=\left(4-x\right)\\ PQ=yandPL=\frac{y}{2}\\ In\Delta OLP,\\ O{L}^2=O{P}^2-L{P}^2=5^2-{\left(\frac{y}{2}\right)}^2\\ \Rightarrow x^2=25-\frac{y^2}{4}-----\left(1\right)\\ In\Delta O^{\prime }LP,\\ O^{\prime }{L}^2=\left(3^2-\frac{y^2}{4}\right)\\ {\left(4-x\right)}^2=9-\frac{y^2}{4}-----\left(2\right)\\ from\left(1\right)and\left(2\right)weget\\ x^2-16+8x-x^2=25-\frac{y^2}{4}-9+\frac{y^2}{4}\\ \Rightarrow -16+8x=16\\ \Rightarrow 8x=32\\ \therefore x=4\\ fromeq{u}^n\left(2\right)\\ 16=25-\frac{y^2}{4}\\ \Rightarrow -9=-\frac{y^2}{4}\\ \Rightarrow y^2=36\\ \therefore y=6cm\end{array}$