Question

1. find the vertex. given, focus (6,-5) and directrix x=0 vertex:

Explanation:

Step1: Recall the property of parabola vertex

The vertex of a parabola is the mid - point between the focus and the directrix.

Step2: Calculate the x - coordinate of the vertex

The formula for the mid - point of two points \(x_1\) (from the focus) and \(x_2\) (from the directrix) is \(x=\frac{x_1 + x_2}{2}\). Given \(x_1 = 6\) (x - coordinate of the focus \((6,-5)\)) and \(x_2=0\) (from the directrix \(x = 0\)), then \(x=\frac{6 + 0}{2}=3\).

Step3: Determine the y - coordinate of the vertex

Since the parabola is symmetric about the axis perpendicular to the directrix. The directrix is \(x = 0\) (a vertical line), and the axis of symmetry is horizontal. The y - coordinate of the vertex is the same as the y - coordinate of the focus. So \(y=-5\).

Answer:

\((3,-5)\)