write an equation for the set of all points in the plane equidistant from (25/4, -6) and x = -1/4. write…

Explanation:

Step1: Find the vertex of the parabola

The vertex ((h,k)) of a parabola is the mid - point between the focus ((x_f,y_f)) and the directrix (x = x_d). For a focus ((x_f,y_f)=\left(\frac{25}{4},-6\right)) and directrix (x =-\frac{1}{4}), the (x) - coordinate of the vertex (h=\frac{\frac{25}{4}+\left(-\frac{1}{4}\right)}{2}=\frac{\frac{25 - 1}{4}}{2}=\frac{6}{2}=3), and (y) - coordinate (k=-6).

Step2: Find the value of (p)

The distance (p) from the vertex to the focus (or from the vertex to the directrix). (p=\frac{25}{4}-3=\frac{25 - 12}{4}=\frac{13}{4})

Step3: Use the vertex form of the parabola equation

The vertex form of a parabola that opens to the right or left is ((y - k)^2=4p(x - h)) Substitute (h = 3), (k=-6), and (p=\frac{13}{4}) into the equation: ((y+6)^2=4\times\frac{13}{4}(x - 3))

Answer:

((y + 6)^2=13(x - 3))