Parabolas Conic Sections

Question

find the vertex and focus of parabola:

x²-10x-12y-23=0

Answer 1

x² - 10x - 12y - 23 = 0

The simplest "vertical" parabola is y = x²

If the vertex is offset, the the more general form is (y - B) = K*(x - A)²

This puts the vertex at (A, B). K is a scaling constant

Rearrange the given equation:

x² - 10x - 12y - 23 = 0

- 12y - 23 = x² - 10x

Add 25 to "complete the square" on the x side:

- 12y - 23 + 25 = x² - 10x + 25

Simplify:

-12y + 2 = (x - 5)²

Divide both sides by -12

(y - 1/6) = -1/12 * (x - 5)²

To complete the problem, look at the discussion here:

https://www.mathsisfun.com/geometry/parabola.html

Answer 2

The standard form of the equation of a parabola (x-h)^2= 4p(y-k)

the vertex is (h,k)

p does NOT equal zero

focus: (h,k+p)

directrix is the line y= k-p

The x^2 tells you it is a up or down parabola

y^2 would be a sideways parabola

Given: x²-10x-12y-23=0

1) Set y to one side: y = 1/12 (x^2-10x-23)

2) Create a T-Chart

y = 1/12 (x^2-10x-23)

x Values Calculate (x,y)

-50 1/12 ((-50)^2-(-50)-23) (-50, 2977/12)

-20 1/12 ((-20)^2-(-20)-23) (-20, 577/12)

-10 1/12 ((-10)^2-(-10)-23) (-10, 59/4)

-3 1/12 ((-3)^2-(-3)-23) (-3, 4/3)

-2 1/12 ((-2)^2-(-2)-23) (-2, 1/12)

-1 (1/12 ((-1)^2-(-1)-23) (-1, -1)

0 1/12 ((0)^2-(0)-23) (0, -23/12)

1 1/12 ((1)^2-(1)-23) (1, -8/3)

2 1/12 ((2)^2-(2)-23) (2, -13/4)

3 1/12 ((3)^2-(3)-23) (3,-11/3)

10 1/12 ((10)^2-(10)-23) (10, -23/12)

20 1/12 ((20)^2-(20)-23) (20, 59/4)

50 1/12 ((50)^2-(50)-23) (50, 659/4)

Now you can graph and see what you are dealing with.

To convert the equation into conics form and find the exact vertex you need to convert the equation to perfect-square

To do this we can use the (b/2)^2 formula to create a new term and complete the square

y = 1/12 (x^2-10x-23) --> (-10/2)^2 = 25 ---> sqrt(25) = 5

y = 1/12 (x-5)^2 -4

(x-h)^2= 4p(y-k) Remember this formula

12(y+4) = x-5^2

(h,k) vertex : ( 5, -4)

Using the vertex form and our graph or (h, k+1/4a)

y=a(x-h)2+k

y = 1/12 (x-5)^2 -4

h= 5 ( two negatives)

k=-4

a=1/12

-4+(1/(4*1/12)) = -1

Focus: (5, -1)