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.