Question

A chord AB of a circle at a distance 6 cm from the centre of a circle whose radius is 6cm less than the chord AB then find the length of the chord and radius of circle.

Answer 1

Let the length of the chord AB = x cm.
Then, radius = (x$-$6) cm.
Since, perpendicular drawn from the centre bisects a chord,
AM = $\frac{\mathrm{AB}}{2}=\frac{x}{2}\mathrm{cm}$

$$\begin{gathered} \mathrm{Now}\mathrm{in}\mathrm{right}△\mathrm{AMO}, \\ {\mathrm{OA}}^2={\mathrm{AM}}^2+{\mathrm{MO}}^2 \\ \Rightarrow {\left(x-6\right)}^2={\left(\frac{x}{2}\right)}^2+6^2 \\ \Rightarrow x^2+36-12x=\frac{x^2}{4}+36 \\ \Rightarrow x^2-\frac{x^2}{4}-12x=0 \\ \Rightarrow 3x^2-48x=0 \\ \Rightarrow 3x(x-16)=0 \\ \Rightarrow \mathit{}x=0\mathrm{or}x=16 \\ \mathrm{But}x\mathrm{cannot}\mathrm{be}0. \\ \therefore \mathrm{Length}\mathrm{of}\text{the}\mathrm{chord}=16\mathrm{cm} \\ \mathrm{and}\mathrm{radius}=x-6=16-6=10\mathrm{cm}. \end{gathered}$$