question_answer

In the parabola \[{{y}^{2}}=4ax,\]the length of the  chord passing through the vertex inclined to the axis at\[\frac{\pi }{4}\] is                   

A)  \[4a\sqrt{2}\]

B)  \[2a\sqrt{2}\]

C)  \[a\sqrt{2}\]

D)  \[a\]

Correct Answer: A

Solution :

Equation of line which is inclined to the axis at \[\frac{\pi }{4}\] is \[y=x\] ?..(i) and equation of parabola is \[{{y}^{2}}=4ax\] ?..(ii) From Eqs. (i) and (ii), we get \[{{x}^{2}}-4ax=0\] \[\Rightarrow \] \[x(x-4a)=0\] \[\Rightarrow \] \[x=0\] or \[x=4a\] \[\therefore \]  If \[x=0,\] then \[y=0\] and if \[x=4a,\] then \[y=4a\] \[\therefore \]  Length of the chord OB is \[|OB|=\sqrt{{{(4a-0)}^{2}}+{{(4a-0)}^{2}}}\] \[=\sqrt{16{{a}^{2}}+16{{a}^{2}}}=4a\sqrt{2}\]