Question

Find the equation of the parabola whose vertex is \[O\left( {0,0} \right)\]and focus is at \[\left( { - 4,0} \right)\].also find equation of directrix . Find the equation of the parabola with vertex at origin and \[y + 3 = 0\] as its directrix. Also find the focus.

Answer 1

Hint:A parabola is a curve where any point is at an equal distance from:
\[ \bullet \] a fixed point (the focus), and
\[ \bullet \] a fixed straight line (the directrix).

Here are the important names:
\[ \bullet \] The directrix and focus (explained above).
\[ \bullet \] The axis of symmetry (goes through the focus, at right angles to the directrix).
\[ \bullet \]The vertex (where the parabola makes its sharpest turn) is half way between the focus and directrix.
\[ \bullet \] The line through the focus parallel to the directrix is the latus rectum (straight side).
\[ \bullet \] Equation of parabola.
\[ \bullet \] Simplest equation of parabola is \[y = {x^2}\]
\[ \bullet \] and another form
\[ \bullet \]\[{y^2} = 4ax\]
Where ‘a’ is the distance from the origin to the focus (and also from the origin to directrix.)

Complete step by step answer:
Given.
Vertex = \[\left( {0,0} \right)\]
Focus = \[\left( { - 4,0} \right)\]

We know that Equation parabola
\[{y^2} = - 4x\]
Where ‘a’ = distance from focus to the origin.
= \[ - a = - 4\] [negative distance from origin (-a)]
= \[a = 4\]
= \[{y^2} = - 4 \times \left( { + 4} \right)x\]
= \[{y^2} = - 16x\]
Required equation of parabola with vertex \[\left( {0,0} \right)\]focus \[\left( { - 4,0} \right)\]
Now, vertex = \[\left( {0,0} \right)\]
  Directrix = \[y + 3 = 0\]
\[y = - 3\]
Therefore, directrix \[\left( {0, - 3} \right)\]

= \[ - a = - 3\] [\[\because - a\]= negative distance from origin]
\[a = 3\]
Equation of parabola
\[{X^2} = 4ay\]
\[{X^2} = 4 \times 3 \times y\]
\[{X^2} = 12y\]

Note:
\[ \bullet \] ‘-a’ means (-) shows direction from origin.
\[ \bullet \]For axis of symmetry along to X and Y axis.
\[ \bullet \] It opens to the left if the coefficient of x is (-)
\[ \bullet \] It opens upward if the coefficient of y is (+).
\[ \bullet \]Parabola has this amazing property:
\[ \to \]Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus.
\[ \bullet \]Rotation of parabola about its axis forms a paraboloid.