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 -

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. Let a parabola is passing through (0,1),(-1,3),(3,3) (2,1) The vertex of the parabola is - (A) (1, (1/3)) (B) ((1/3), 1) (C) (1,3) (D) (3,1)

Tardigrade Admin · Accepted Answer

Axis of parabola is bisector of parallel chord A B

& C D

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} (y - \frac{1}{3})

Vertex

(1, \frac{1}{3})

$(1, \frac{1}{3})$

$(\frac{1}{3}, 1)$

$(1, 3)$

$(3, 1)$

(1,31​)

(31​,1)

(1,3)

(3,1)

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=ay+b It passing (0,1)&(3,3) so 1=a+b...(1) 4=3a+b...(2) from (1) & (2) a=23​&b=−21​ (x−1)2=23​(y−31​) Vertex (1,31​)

$(1, \frac{1}{3})$

$(\frac{1}{3}, 1)$

$(1, 3)$

$(3, 1)$