Question

the focus of a parabola is (5, 5), and its directrix is y = 3. the point (x, -7) is on the parabola. what is the distance between this point and the focus?
units

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of points equidistant from the focus and the directrix. So, the distance from the point \((x, -7)\) to the focus \((5, 5)\) is equal to the distance from the point \((x, -7)\) to the directrix \(y = 3\).

Step2: Calculate the distance from the point to the directrix

The distance from a point \((x_0, y_0)\) to the line \(y = k\) is \(|y_0 - k|\). Here, \(x_0=x\), \(y_0=-7\), and \(k = 3\). So the distance is \(|-7 - 3|=|-10| = 10\).

Step3: Conclude the distance to the focus

Since the point is on the parabola, the distance to the focus is equal to the distance to the directrix. So the distance between \((x, -7)\) and \((5, 5)\) is \(10\).

Answer:

10