5.1 · Circles and parabolas: completing the square and geometry

Complete the square without changing the equation

A circle has form $\displaystyle (x-h)^2+(y-k)^2=r^2$, with center $\displaystyle (h,k)$ and radius $\displaystyle r>0$. When completing squares, add the required amounts to both sides.

For $\displaystyle x^2+y^2-4x+6y-12=0$, completion yields $\displaystyle (x-2)^2+(y+3)^2=25$, revealing center $\displaystyle (2,-3)$ and radius $\displaystyle 5$.

A parabola is an equal-distance locus

A point on a parabola is equally distant from its focus and directrix. In $\displaystyle (x-h)^2=4p(y-k)$, the vertex is $\displaystyle (h,k)$, focus is $\displaystyle (h,k+p)$, and directrix is $\displaystyle y=k-p$.

For $\displaystyle (y-k)^2=4p(x-h)$, the opening is horizontal. The sign of $\displaystyle p$ determines up versus down or right versus left. The coefficient is $\displaystyle 4p$, not $\displaystyle p$ itself.

$$(x-h)^2=4p(y-k)$$

Use a diagram to verify the features

For $\displaystyle y^2=8x$, the vertex is the origin, the focus is $\displaystyle (2,0)$, and directrix is $\displaystyle x=-2$. A horizontal axis of symmetry matches the horizontal displacement of the focus.

Check the original equation at a simple point and compare its distances to the focus and directrix. This confirms the geometry rather than just recalling a formula.