Question

The axis of the parabola \[{x^2} - 4x - y + 1 = 0\] is

Answer 1

Hint:
 In the given problem we are provided with a parabola whose axis we have to find. So to find the axis we will start by simplifying the given problem. Then we can easily use the vertex form of our simplified equation, we will reach our needed result.

Complete step by step solution:
We are given,
\[{x^2} - 4x - y + 1 = 0\]
On solving for y we get,
\[ \Rightarrow y = {x^2} - 4x + 1\]
On splitting the last term on left hand side, we get,
\[ \Rightarrow y = {x^2} - 4x + 4 - 3\]
Using \[{a^2} - 2ab + {b^2} = {(a - b)^2}\], we get,
\[ \Rightarrow y = {(x - 2)^2} - 3\]
On adding 3 on both sides we get,
\[ \Rightarrow y + 3 = {(x - 2)^2}\]

Now, if we have, the vertex form of the equation, we get, our Axis of the parabola as,
\[x - 2 = 0\]

Note:
Here are some given properties of a parabola,
1) The eccentricity of any parabola is 1.
2) The parabola is symmetric about its axis.
3) The axis is perpendicular to the directrix.
4) The axis passes through the vertex and the focus.
5) The tangent at the vertex is parallel to the directrix.
6) The vertex is the midpoint of the focus and the point of intersection of directrix and axis.
7) Tangents drawn to any point on the directrix are perpendicular.