Find the vertex and latus rectum of parabola, whose axis is parallel to Y - axis and which passes through the points (0, 2), (1, 3) and (-2, 6), is

Find the vertex and latus rectum of parabola, whose axis is parallel to Y - axis and which passes through the points (0, 2), (1, 3) and (-2, 6), is Correct Answer Vertex = (0, 2) and latus rectum = 1

Concept:

Equation of parabola along Y-axis: (x – h)² = ±4a (y - k),

Here, point (h, k) is vertex of parabola, 4a = length of latus rectum and focus (0, ±a)

Calculation:

Let the vertex of the parabola be (h, k) 

And length of its latus rectum be 4a.

Since its axis is parallel to y - axis,

Its equation can be written as

(x − h)² = 4a(y − k)     ----(1)

It passes through the points (0, 2), (1, 3) and (-2, 6)

So, For point (0, 2)

(0 − h)² = 4a(2− k)

⇒ h² = 4a(2 − k)      ----(2)

And for point (1, 3)

(1 − h)² = 4a(3 − k) 

⇒ 1 − 2h + h² = 4a(3 − k)      ----(3)

And for point (-2, 6)

(-2 − h)² = 4a(6 − k) 

⇒ 4 + 4h + h² = 4a(6 − k)     ----(4)

Subtracting (2), (3) and (3), (4) we get

1 − 2h = 4a     ----(5)

And 3 + 6h = 12a 

i.e. 1 + 2h = 4a     ----(6)

Then solving (5) and (6), we get

a = 1/4​ and h = 0

From equation (2), we get

k = 2​.

So, Vertex = (h, k) = (0, 2) and

Latus rectum = 4a = 4(1/4) = 1

Hence, option (3) is correct.