Question

a parabola has focus (-4, 1) and directrix given by the equation y = 5. complete the sentence. the point a(-2, 0) lies the parabola because the distance between a and the focus is the distance between a and the directrix.

Explanation:

Step1: Calculate the distance between \(A(-2,0)\) and the focus \((-4,1)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here, \(x_1=-2,y_1 = 0,x_2=-4,y_2 = 1\).
\(d_1=\sqrt{(-4+2)^2+(1 - 0)^2}=\sqrt{(-2)^2+1^2}=\sqrt{4 + 1}=\sqrt{5}\).

Step2: Calculate the distance between \(A(-2,0)\) and the directrix \(y = 5\)

The distance \(d_2\) from a point \((x_0,y_0)\) to the line \(y = k\) is \(|y_0 - k|\).
Here, \(x_0=-2,y_0 = 0,k = 5\).
\(d_2=|0 - 5|=5\).

Answer:

The point \(A(-2,0)\) lies outside the parabola because the distance between \(A\) and the focus is less than the distance between \(A\) and the directrix.