If two parallel chords of a circle, having diameter 4 units, lie on opposite sides of the …

Question

If two parallel chords of a circle, having diameter $4$ units, lie on opposite sides of the centre and subtend angles $\cos^{-1}\dfrac17$ and $\sec^{-1}(7)$ at the centre respectively, then the distance between these chords is:

Accepted Answer

$r=2$. For the angle $2\theta$ with $\cos2\theta=\dfrac17$: $\cos^2\theta=\dfrac{1+1/7}2=\dfrac47$.

For the angle $2\phi$ with $\sec2\phi=7\Rightarrow\cos2\phi=\dfrac17$ also, giving the same $\cos^2\phi=\dfrac47$... using the complementary relation, $\cos\phi=\dfrac2{\sqrt7}$.

Distance from centre to each chord: $r\cos\theta$ and $r\cos\phi$, both $=\dfrac4{\sqrt7}$.

$$\text{Total distance}=\frac4{\sqrt7}+\frac4{\sqrt7}=\frac8{\sqrt7}.$$

Solutions · 1