Answer
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Hint: Here first we will observe that the x coordinate of vertex is zero. So, the axis of the parabola should be $y$ axis. Next it is given that the vertex is above focus, so it is an upward parabola which is cutting the y axis at \[\left( {0,\,2} \right)\]. Therefore, if we represent the equation of parabola as $y = a{\left( {x - h} \right)^2} + k$, then h should be zero here and a should be negative.
Complete step by step answer:
In the above question, it is given that there is a parabola whose vertex is \[\left( {0,\,2} \right)\] and the vertex is above focus. Therefore, we can represent the above parabola as,
We can write a parabola in its standard form as $y = a{\left( {x - h} \right)^2} + k$, where $x - h = 0$ is the axis of symmetry and $\left( {h,k} \right)$ is the vertex. So, basically there are four things to note here:
- Axis of symmetry is perpendicular to the directrix and passes through the focus and hence the axis of symmetry is $x = 0$.
- Hence, the axis of symmetry and the directrix cut each other at $\left( {0,4} \right)$.
- Vertex is always midway between the directrix and the focus and hence the vertex is $\left( {0,2} \right)$.
- As vertex (and directrix) is above focus, parabola will have a maxima at vertex and a is negative.
Hence, we can write the standard form of equation of parabola as $y = a{\left( {x - 0} \right)^2} + 2$ or $y = a{\left( x \right)^2} + 2$ , where $a < 0$.
Note:Here we should be careful while deciding that it is an upward parabola or a downward parabola. If the vertex is above the focus, then it is an upward parabola and if the vertex is below the focus, then it is a downward parabola. Also, if some constant term is subtracted or added in the equation, then its vertex should not be the origin. There should be some other point.
Complete step by step answer:
In the above question, it is given that there is a parabola whose vertex is \[\left( {0,\,2} \right)\] and the vertex is above focus. Therefore, we can represent the above parabola as,
We can write a parabola in its standard form as $y = a{\left( {x - h} \right)^2} + k$, where $x - h = 0$ is the axis of symmetry and $\left( {h,k} \right)$ is the vertex. So, basically there are four things to note here:
- Axis of symmetry is perpendicular to the directrix and passes through the focus and hence the axis of symmetry is $x = 0$.
- Hence, the axis of symmetry and the directrix cut each other at $\left( {0,4} \right)$.
- Vertex is always midway between the directrix and the focus and hence the vertex is $\left( {0,2} \right)$.
- As vertex (and directrix) is above focus, parabola will have a maxima at vertex and a is negative.
Hence, we can write the standard form of equation of parabola as $y = a{\left( {x - 0} \right)^2} + 2$ or $y = a{\left( x \right)^2} + 2$ , where $a < 0$.
Note:Here we should be careful while deciding that it is an upward parabola or a downward parabola. If the vertex is above the focus, then it is an upward parabola and if the vertex is below the focus, then it is a downward parabola. Also, if some constant term is subtracted or added in the equation, then its vertex should not be the origin. There should be some other point.
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