Question

1. find the vertex. given, focus (2,4) and directrix y = -6 (hint: vertex is in the middle of focus and directrix.)

Explanation:

Step1: Find the x - coordinate of the vertex

Since the vertex is in the middle of the focus and the directrix. For a parabola, if the focus is \((x_f,y_f)=(2,4)\) and the directrix is \(y = - 6\), the x - coordinate of the vertex is the same as the x - coordinate of the focus. So \(x = 2\).

Step2: Find the y - coordinate of the vertex

Use the mid - point formula for the y - values. The mid - point formula between two points \(y_1\) and \(y_2\) is \(y=\frac{y_1 + y_2}{2}\). Here \(y_1 = 4\) (y - coordinate of the focus) and \(y_2=-6\) (y - value of the directrix). Then \(y=\frac{4+( - 6)}{2}=\frac{4 - 6}{2}=\frac{-2}{2}=-1\).

Answer:

\((2,-1)\)