Question

Rewrite as y−k=a(x−h)2 or x−h=a(y−k)2. Find the vertex, focus, and directrix. y−3=(2−x)^2

Answer 1

Given:

The given equation is

$y-3=(2-x)^2$

To find:

The vertex, focus, and directrix.

Solution:

The equation of a parabola is

$y-k=a(x-h)^2$     ...(i)

where, (h,k) is vertex, $\left(h,k+\dfrac{1}{4a}\right)$ and directrix is $y=k-\dfrac{1}{4a}$

We have,

$y-3=(2-x)^2$

It can be written as

$y-3=(-(x-2))^2$

$y-3=(x-2)^2$    ...(ii)

On comparing (i) and (ii), we get

$h=2,k=3,a=1$

Vertex of the parabola is (2,3).

$Focus=\left(2,3+\dfrac{1}{4(1)}\right)$

$Focus=\left(2,3+\dfrac{1}{4}\right)$

$Focus=\left(2,\dfrac{13}{4}\right)$

Therefore, the focus of the parabola is $\left(2,\dfrac{13}{4}\right)$.

Directrix of the parabola is

$y=3-\dfrac{1}{4(1)}$

$y=3-\dfrac{1}{4}$

$y=\dfrac{11}{4}$

Therefore, the directrix of the parabola is $y=\dfrac{11}{4}$.