question_answer

The condition that the line \[x\cos \alpha +y\sin \alpha =p\] may touch the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is                                             [AMU 1999]

A)            \[p=a\cos \alpha \]                  

B)            \[p=a\tan \alpha \]

C)            \[{{p}^{2}}={{a}^{2}}\] 

D)            \[p\sin \alpha =a\]

Correct Answer: C

Solution :

           According to question length of perpendicular on the line from the centre of circle is equal to the radius of the circle.                    OM = radius of circle                    \[\left| \frac{-p}{\sqrt{{{\cos }^{2}}\alpha +{{\sin }^{2}}\alpha }} \right|=a\]Þ \[|-p|\ =a\Rightarrow {{p}^{2}}={{a}^{2}}\].