Question

A parabola can be drawn given a focus of left parenthesis, 4, comma, 7, right parenthesis(4,7) and a directrix of x, equals, minus, 4x=−4. What can be said about the parabola? Include p-value

Answer 1

The directrix x = −4 is vertical, so the axis is horizontal through the focus y = 7. The vertex is midway in x between the focus x = 4 and the directrix x = −4, so the vertex is (0,7). The focal distance p (vertex to focus) is 4, so the parabola opens to the right.

Standard form: (y − 7)^2 = 4p(x − 0) = 16x.

Other facts: axis y = 7, focus (4,7), directrix x = −4, latus rectum length = 4p = 16 (endpoints at (4, 7 ± 8)).