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