question_answer

Find the length of the line segment joining the vertex of the parabola \[{{y}^{2}}=4ax\] and a point on the parabola where the line segment makes an angle \['\theta '\] to the x-axis.

A) \[\frac{2a\cos \theta }{{{\sin }^{2}}\theta }\]

B) \[\frac{4a\cos \theta }{{{\sin }^{2}}\theta }\]

C) \[\frac{4a\cos \theta }{3{{\sin }^{2}}\theta }\]

D)  None of the above

Correct Answer: A

Solution :

Let any point \[(h,\,\,k)\] will satisfy                 \[{{y}^{2}}=4ax\]  \[i.e.,\]   \[{{k}^{2}}=4ah\]       ... (i) Let a line \[OP\] makes an angle \[\theta \] from the axis. \[\therefore \]  In\[\Delta OAP,\,\,\sin \theta =\frac{PA}{OP}\] \[\Rightarrow \]               \[\sin \theta =\frac{k}{l}\]            \[\Rightarrow \]               \[k=l\sin \theta \] and        \[\cos \theta =\frac{OA}{OP}\]  \[\Rightarrow \]               \[\cos \theta =\frac{h}{l}\] From Eq. (i), we get \[{{l}^{2}}{{\sin }^{2}}\theta =4a\times l\cos \theta \]                                 (put\[k=l\sin \theta ,\,\,h=l\cos \theta )\] \[\Rightarrow \]               \[l=\frac{4a\cos \theta }{{{\sin }^{2}}\theta }\]