Questions: Find the coordinates of the vertex and the focus for the parabola given by the following equation: x^2-2x-8y-63=0 Round all answers to 2 places after the decimal point, if necessary. Coordinates of the vertex: ( , ) Coordinates of the focus: , )

Find the coordinates of the vertex and the focus for the parabola given by the following equation:
x^2-2x-8y-63=0

Round all answers to 2 places after the decimal point, if necessary.
Coordinates of the vertex: ( , )

Coordinates of the focus: , )

Transcript text: Find the coordinates of the vertex and the focus for the parabola given by the following equation: \[ x^{2}-2 x-8 y-63=0 \] Round all answers to 2 places after the decimal point, if necessary. Coordinates of the vertex: ( $\square$ , $\square$ ) Coordinates of the focus: $\square$ , $\square$ )

Solution

Solution Steps

Step 1: Rewrite the equation in standard form

Start by rewriting the given equation: \[ x^{2} - 2x - 8y - 63 = 0 \] Move the $ y $-terms and constants to the other side: \[ x^{2} - 2x = 8y + 63 \]

Step 2: Complete the square for the $ x $-terms

To complete the square for $ x^{2} - 2x $, take half of the coefficient of $ x $, square it, and add it to both sides: \[ x^{2} - 2x + 1 = 8y + 63 + 1 \] This simplifies to: \[ (x - 1)^{2} = 8y + 64 \]

Step 3: Rewrite in vertex form

Divide both sides by 8 to isolate $ y $: \[ \frac{(x - 1)^{2}}{8} = y + 8 \] Rearrange to write in vertex form: \[ y = \frac{1}{8}(x - 1)^{2} - 8 \] From this, the vertex is at $ (1, -8) $.

Step 4: Identify the focus

The standard form of a parabola is $ y = \frac{1}{4p}(x - h)^{2} + k $, where $ (h, k) $ is the vertex and $ p $ is the distance from the vertex to the focus. Comparing: \[ \frac{1}{4p} = \frac{1}{8} \implies p = 2 \] Since the parabola opens upwards, the focus is at $ (1, -8 + 2) = (1, -6) $.

Step 5: Round the coordinates

Round the coordinates to two decimal places: