question_answer

If the chord \[y=mx+1\] of the circle \[{{x}^{2}}+{{y}^{2}}=1\] subtends an angle of measure \[{{45}^{o}}\] at the major segment of the circle then value of m is                                                  [AIEEE 2002]

A)            2    

B)            - 2

C)            - 1  

D)            None of these

Correct Answer: C

Solution :

           Given circle is \[{{x}^{2}}+{{y}^{2}}=1\]                          \[C(0,0)\] and radius = 1 and chord is \[y=mx+1\] \[\cos {{45}^{o}}=\frac{CP}{CR}\]                    \[CP=\]Perpendicular distance from (0,0) to chord \[y=mx+1\]                    \[CP=\]\[\frac{1}{\sqrt{{{m}^{2}}+1}}\]              (CR = radius = 1)                    \[\cos 45{}^\circ =\frac{1/\sqrt{{{m}^{2}}+1}}{1}\Rightarrow \frac{1}{\sqrt{2}}=\frac{1}{\sqrt{{{m}^{2}}+1}}\]                    \[{{m}^{2}}+1=2\] Þ \[m=\pm 1.\]