question_answer

    If M is the foot of the perpendicular from a point P oh a parabola to its directrix and SPM is an equilateral triangle, where\[S\]is the focus, then PM is equal to

A)  \[a\]                                    

B)  \[2a\]

C)  \[3a\]                                  

D)  \[4a\]

Correct Answer: D

Solution :

                Since,\[\Delta SPM\]is an equilateral triangle. Therefore, \[SP=PM=SM\] \[\therefore \]                  \[\angle PMZ=90{}^\circ \] and        \[\angle PMS=60{}^\circ \] \[\therefore \]                  \[\angle SMZ=30{}^\circ \] Now, in right angle\[\Delta SMZ\]                 \[\sin 30{}^\circ =\frac{SZ}{SM}\] \[\Rightarrow \]               \[\frac{1}{2}=\frac{2a}{SM}\] \[\Rightarrow \]               \[SM=4a\] \[\Rightarrow \]               \[PM=4a\]               \[(\because PM=SM)\]