Question

Observe the following facts for a parabola :
(i) Axis of the parabola is the only line which can be the perpendicular bisector of the two chords of the parabola.
(ii) If $AB$ and $CD$ are two parallel chords of the parabola and the normals at $A$ and $B$ intersect at $P$ and the normals at $C$ and $D$ intersect at $Q$, then $PQ$ is a normal to the parabola.
The directrix of the parabola is -

• A. $y-\frac{1}{24}=0$
• B. $y+\frac{1}{2}=0$
• C. $y+\frac{1}{24}=0$
• D. $y+\frac{1}{12}=0$

Answer 1

Answer: (C)

Axis of parabola is bisector of parallel chord A B $\& CD$ are parallel chord.
so axis $x =1$

equation of parabola is
$(x-1)^2=a y+b$
It passing $(0,1) \&(3,3)$
so $1= a + b$...(1)
$4=3 a+b$...(2)
from (1) & (2)
$a=\frac{3}{2} \& b=-\frac{1}{2}$
$(x-1)^2=\frac{3}{2}\left(y-\frac{1}{3}\right)$
$a =\frac{3}{8}$
directrix of $x^2=4 a y$ is $y=-a$
$y-\frac{1}{3}=-\frac{3}{8} $
$\Rightarrow y=\frac{1}{3}-\frac{3}{8} $
$y+\frac{1}{24}=0$