Derive the equation of the parabola with a focus at (6, 2) and a directrix of y = 1.
f(x) = −1/2 times (x − 6)^2 + 3/2
f(x) = 1/2 times (x − 6)^2 + 3/2
f(x) = −1/2 times (x + 3/2)^2 + 6
f(x) = 1/2 times (x + 3/2)^2 + 6

f(x) = 1/2 times (x − 6)^2 + 3/2

Step-by-step explanation:

The definition of a parabola states that any point of the parabola is at the same distance from the focus and from the directrix.  Taking focus at (h, k) and a directrix at y = d:

distance between point (x, y) and the focus = sqrt((x - h)^2 + (y - k)^2)

distance between point (x, y) and the directrix = sqrt((y - d)^2)

Equating them and eliminating the square root:

(x -h)^2 + (y - k)^2 = (y - d)^2

Expanding the y-terms and isolating y

(x -h)^2 + y^2 - 2yk + k^2 = y^2 - 2yd + d^2

(x -h)^2 + k^2 - d^2 = 2y(k - d)

(x -h)^2 + (k - d)(k + d) = 2y(k - d)

(x -h)^2/(2(k - d)) + (k + d)/2 = y

Here h =6, k = 2, d = 1 and y = f(x). Replacing

f(x) = (x -6)^2/(2(2 - 1)) + (2 + 1)/2

f(x) = (x -6)^2/2 + 3/2

d. 24+-0.114

step-by-step explanation:

the question requires us to calculate the approximate 95% confidence interval for the mean number of ounces of ketchup per bottle in the sample given the following statistics;

sample mean = 24

standard deviation, σ = 0.4

sample size , n = 49

the confidence interval for a population mean is calculated using the formula;

sample mean ± z-score*(σ/sqrt(n))

the z-score associated with a 95% confidence interval, from the standard normal tables, is;

1.96

substituting these values into the formula;

24 ± 1.96*(0.4/7)

= 24 ± 0.112

from the alternatives given, choice d is closest to the above expression;