Question

Find the equation of the parabola having the vertex at $\left( {0,1} \right)$ and the focus at $\left( {0,0} \right)$ :
A. ${x^2} + 4y - 4 = 0$
B. ${x^2} + 4y + 4 = 0$
C. ${x^2} - 4y + 4 = 0$
D. ${x^2} - 4y - 4 = 0$

Answer 1

Hint : Take the general equation of the parabola and then substitute the values of the vertex. Calculate the value of $a$ by taking the distance of the vertex from the focus and then substitute in the equation of parabola.

Complete step-by-step answer :
As given in the question, the parabola has the vertex at $\left( {0,1} \right)$ and the focus at $\left( {0,0} \right)$ . The focus lies below the vertex. So, the general equation of the parabola is ${\left( {x - h} \right)^2} = - 4a\left( {y - k} \right)$ where $\left( {h,k} \right)$ is the vertex of the parabola and $a$ is the distance between the vertex and the focus.
Substitute $0$ for $h$ and $1$ for $k$ in the equation of parabola as per given in the question:
${\left( {x - h} \right)^2} = - 4a\left( {y - k} \right) \\\ {\left( {x - 0} \right)^2} = - 4a\left( {y - 1} \right) \\\ {x^2} = - 4a\left( {y - 1} \right)\;\;\;\;\;\;\; \ldots \left( 1 \right) \;$
Now calculate the distance between the vertex $\left( {0,1} \right)$ and focus $\left( {0,0} \right)$ by using the distance formula.
$\sqrt {{{\left( {0 - 0} \right)}^2} + {{\left( {0 - 1} \right)}^2}} = \sqrt {{0^2} + {{\left( { - 1} \right)}^2}} \\\ = \sqrt 1 \\\ = 1 \;$
As the value of $a$ is the distance of the focus from the vertex which is equal to $1$ .
Substitute $1$ for $a$ in the equation $\left( 1 \right)$ of parabola.
${x^2} = - 4a\left( {y - 1} \right) \\\ {x^2} = - 4\left( 1 \right)\left( {y - 1} \right) \\\ {x^2} = - 4y + 4 \\\ {x^2} + 4y - 4 = 0 \;$
So, the equation of the parabola is equal to ${x^2} + 4y - 4 = 0$ .
So, the correct answer is “Option A”.

Note : The general equation of the parabola with focus below the vertex is equal to ${\left( {x - h} \right)^2} = - 4a\left( {y - k} \right)$ where $\left( {h,k} \right)$ is the vertex of the parabola and $a$ is the distance between the vertex and the focus. The distance between two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is equal to $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$ with the help of the distance formula in two dimensional geometry.