6.0 The generalized form of parabola: ${\left( {y - k} \right)^2} = 4a\left( {x - h} \right)$

The standard equation of parabola $${y^2} = 4ax...(1)$$

can be written as $${\left( {y - 0} \right)^2} = 4a\left( {x - 0} \right)$$

As shown in figure $(12)$, the vertex of the parabola is $A(0,0)$. When the origin is shifted from $A(0,0)$ to $A'(h,k)$

without changing the direction of axes, the equation of parabola becomes, $${\left( {y - k} \right)^2} = 4a\left( {x - h} \right)...(2)$$

The equation $(2)$ is called the generalized form of parabola given in equation $(1)$. The axis $A'X''\parallel AX$ with its vertex at $A'(h,k)$ and focus at $(a+h,k)$ having length of latus rectum$=4a$ and the equation of directrix is $$x+a-h=0$$

Similarly for another form of parabola $${\left( {x - h} \right)^2} = 4a\left( {y - k} \right)$$ axis parallel to $Y$-axis with its vertex $(h,k)$, focus $(h,a+k)$ having length of latus rectum$=4a$ and the equation of directrix is $$y+a-k=0$$