Question

jessica remembers that for any point (x, y) on the parabola, the distance to the focus is the same as the distance to the directrix. she sets these two distances equal to each other. which equation is the result of this?
$y + 1 = \sqrt{x^2 + (y - 2)^2}$
$y - 1 = \sqrt{x^2 + (y - 2)^2}$
$y + 1 = \sqrt{x^2 + y^2}$

Explanation:

Step1: Identify Focus and Directrix

From the graph, the focus is at \((0, 2)\) (the orange dot on the y - axis) and the directrix is the horizontal line \(y=- 1\) (the dashed orange line).

Step2: Distance to Focus

The distance between a point \((x,y)\) on the parabola and the focus \((0,2)\) is given by the distance formula \(d=\sqrt{(x - x_1)^2+(y - y_1)^2}\), where \((x_1,y_1)=(0,2)\). So the distance to the focus is \(\sqrt{(x - 0)^2+(y - 2)^2}=\sqrt{x^{2}+(y - 2)^{2}}\).

Step3: Distance to Directrix

The distance between a point \((x,y)\) and the horizontal line \(y = - 1\) is the vertical distance, which is \(|y-(-1)|=|y + 1|\). Since the parabola opens upwards (from the graph), \(y\geq0\) (the vertex is at \((0,0.5)\) approximately, and the parabola is above the directrix \(y=-1\)), so \(y + 1\) is non - negative and the distance is \(y + 1\).

Step4: Set Distances Equal

Since the distance from \((x,y)\) to the focus equals the distance from \((x,y)\) to the directrix, we set \(y + 1=\sqrt{x^{2}+(y - 2)^{2}}\).

Answer:

\(y + 1=\sqrt{x^{2}+(y - 2)^{2}}\) (the first option)