Question

How do you find the equation of a parabola with vertex at the origin and directrix x=3?

Answer 1

Hint: Start the equation with writing the general equation of the parabola and directrix. After this substitute the given values in the general equation and make use of the general equation of the directrix to further solve the general equation of parabola and you get the desired equation of parabola which satisfies the given conditions.

Complete step by step solution:
WE have been told to find the equation of parabola with vertex at (0, 0) and directrix x=3, where x=3 is a line parallel to y-axis.
Now to find the equation of the parabola we will first try to note down the general equation of the parabola which has an axis of symmetry as y=0.
The general equation is given as ${(y - {\text{k}})^2} = 4{\text{p}}\left( {x - {\text{h}}} \right)$ and directrix is given by $x = {\text{h - p}}$
Since our vertex is at origin, the value of (h, k) = (0, 0)
Therefore we get
$
  \Rightarrow {(y - 0)^2} = 4p(x - 0) \\
   \Rightarrow {y^2} = 4px - - - (1) \\
 $
Also to find the value of p, substitute our value of directrix and value of h=0 in general equation of directrix.
$
  \Rightarrow x = {\text{h - p}} \\
   \Rightarrow 3 = 0 - p \\
   \Rightarrow p = - 3 \\
 $
Hence substituting the value of p= -3 in the equation (1) we get
$
  \Rightarrow {y^2} = 4 \times \left( { - 3} \right)x \\
   \Rightarrow {y^2} = - 12x \\
 $
Solving further we get
${y^2} + 12x = 0$
This is the equation of the circle with centre at origin and directrix x=-3.

Hence the answer is ${y^2} + 12x = 0$ which is the directrix of the parabola

Note: In this solution we have used the simplified general formula for the parabola. If you are not comfortable with using the simplified form you can use the basic general formula of the parabola which goes like ${(x - {\text{p}})^2} + {y^2} = {\left( {x + p} \right)^2}$. This simplified further gives us the equation we have used above.