# A parabola has directrix y = -3, axis x = 4 and latus rectum equal to 2. Find its equations.

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The latus rectum, L= 2.

If the distance between directrix and focus is p = 2a, then L is,

`L = 4a`

Therefore, `2 = 4a` and this gives, `a = 1/2` .

The distance between directrix and focus = 2 times distance between vertex and directrix = 2 times distance between focus and vertex.

Therefore the distance between vertex and directrix = a = 1/2

The line of symmetry is x =4 and the vertex is on this line.

Therefore the coordinates for vertex are,

either `(4,-3+1/2)` or `(4,-3-1/2)`

Vertex is `(4,-5/2)` or `(4,-7/2)` .

If the latus rectum is 4a and `(x_0,y_0)` is the vertex, then the equation is given by,

`4a(y-y_0) = (x-x_0)^2 `

Therefore the equations are,

For `(4,-5/2)` ,

`2(y+5/2) = (x-4)^2`

`y = 1/2(x-4)^2-5/2`

For `(4,-7/2)` ,

`2(y+7/2) = -(x-4)^2 `

(This should be minus, since its the other opposite)

`y = -1/2(x-4)^2-7/2`

Therefore two equations are, `y = 1/2(x-4)^2-5/2` and `y = -1/2(x-4)^2-7/2`.