question_answer

Let C be a point on a straight line AB. Circles are drawn with diameters AC and AB. Let P be any point on the circumference of the circle with diameter AB. If AP meets the other circle at Q, then

A) \[OC\parallel PB\]

B) OC is never parallel of PB

C) \[QC=\frac{1}{2}PB\]

D) \[QC\parallel PB\] and \[QC=\frac{1}{2}PB\]

Correct Answer: A

Solution :

[a] 0 In \[\Delta AQC\]and \[\Delta APB\] \[\angle AQC=\angle APB\] (angles made in semi-circle) \[\angle QAC=\angle PAB\]                     (common) \[\therefore \]      \[\angle ACQ=\angle ABP\] \[\Rightarrow \]   \[\Delta AQC\sim \Delta \Alpha PB\] \[\therefore \]      \[\Delta \frac{AQ}{AP}=\frac{AC}{AB}\]\[\Rightarrow \]\[QC\parallel PB\]