Problem
GEO-B2-M05-P011 Circle with a Diameter
#11
★★★☆☆ Level 3 of 5
An inversion has radius \(6\). A circle has diameter \(OQ\), where \(OQ=8\). Find the distance from \(O\) to the image line of this circle.
Point \(Q\) maps to \(Q'\), and the image line is perpendicular to \(OQ\).
The circle passes through \(O\), so its image is a line. For \(Q'\), \(OQ\cdot OQ'=36\), hence \(OQ'=\frac{36}{8}=\frac{9}{2}\). The image line passes through \(Q'\) perpendicular to \(OQ\), so its distance from \(O\) is \(\frac{9}{2}\).
A good problem on the geometry of the image line.