Q. In the parabola $ {{y}^{2}}=4ax, $ the length of the chord passing through the vertex inclined to the axis at $ \frac{\pi }{4} $ is
J & K CETJ & K CET 2007
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} $
