Question

1. write the equation and sketch the parabola with a focus at (0,1) and a directrix line at y = 9, one point at (8,1) and additional point at (-4,4) (20 points).

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of all points \((x,y)\) such that the distance from the point \((x,y)\) to the focus \((x_f,y_f)=(0,1)\) is equal to the distance from the point \((x,y)\) to the directrix \(y = 9\).

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and the distance from a point \((x,y)\) to the line \(y = k\) is \(|y - k|\).

The distance from \((x,y)\) to the focus \((0,1)\) is \(\sqrt{(x - 0)^2+(y - 1)^2}\), and the distance from \((x,y)\) to the directrix \(y=9\) is \(|y - 9|\).

Step2: Set up the equation

By the definition of a parabola \(\sqrt{x^{2}+(y - 1)^{2}}=|y - 9|\).

Square both sides to get rid of the square - root and absolute value (since \((a=b)\Rightarrow(a^{2}=b^{2})\) for real numbers \(a\) and \(b\)):

\(x^{2}+(y - 1)^{2}=(y - 9)^{2}\)

Expand the squares:

\(x^{2}+y^{2}-2y + 1=y^{2}-18y + 81\)

Subtract \(y^{2}\) from both sides:

\(x^{2}-2y + 1=-18y + 81\)

Add \(18y\) to both sides and subtract \(1\) from both sides:

\(x^{2}+16y-80 = 0\)

Solve for \(y\):

\(16y=-x^{2}+80\)

\(y=-\frac{1}{16}x^{2}+5\)

Step3: Check the given point \((8,1)\)

Substitute \(x = 8\) into the equation \(y=-\frac{1}{16}x^{2}+5\):

\(y=-\frac{1}{16}\times8^{2}+5=-\frac{1}{16}\times64 + 5=-4 + 5=1\)

Step4: Find another point \((-4,4)\)

Substitute \(x=-4\) into the equation \(y=-\frac{1}{16}x^{2}+5\):

\(y=-\frac{1}{16}\times(-4)^{2}+5=-\frac{1}{16}\times16 + 5=-1 + 5=4\)

Answer:

The equation of the parabola is \(y =-\frac{1}{16}x^{2}+5\)