1. Tardigrade
  2. Question
  3. Mathematics
  4. 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. Let a parabola is passing through (0,1),(-1,3),(3,3) (2,1) The vertex of the parabola is -

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Q. 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.
Let a parabola is passing through $(0,1),(-1,3),(3,3)$ & $(2,1)$
The vertex of the parabola is -

Conic Sections

Solution:

Axis of parabola is bisector of parallel chord A B $\& CD$ are parallel chord.
so axis $x =1$
image
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)
$a=\frac{3}{2} \& b=-\frac{1}{2}$
$(x-1)^2=\frac{3}{2}\left(y-\frac{1}{3}\right)$
Vertex $\left(1, \frac{1}{3}\right)$