Write the equation in vertex form for the parabola with focus (7, –27/4) and directrix y= – 13/4

Mathematics

1 answer:

Nataliya [291]2 years ago

7 0

Answer:

x2 +8x +4y +4 = 0

4y= -x2 -8x -4y = -.25*x^2 -2x -1

a = -.25b = -2c = -1

x position of vertex:

h = -b / 2a

h = 2 / 2*-.25h = 2 / -.5h = -4

y position of vertex:

k = ah^2 + bh + ck = -.25*-4^2 + -2*-4 + -1k = -4 +8 -1k = 3

VERTEX = (-4, 3)*

x value of focus =x value of vertex = -4

y value of focus =(1 (-b^2 -4ac)) / 4a

a = -.25 b = -2 c =-1

y value = (1 (-4 -4*-.25*-1)) / 4*-.25

y value = (1 (-4 -4*-.25*-1)) / -1

y value = (1 -4 +1) / -1y value = (-2 / -1)y value = 2

focus value = (-4, 2)

Answer is the last one.

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Step-by-step explanation:

According to the attached sketch, there are 2 triangles which we need to focus on, triangle A (in yellow) and triangle B (In red).

If you look at triangle A, we notice that X is the hypotenuse of triangle A. This means that X must be the largest length in triangle A, hence we can say that x must be greater than 24 (or 24 < x)

Now look at triangle B, in this case, they hypotenuse is 30 and x is the length of one of the sides. This means that x must be shorter than the hypotenuse (i.e x < 30)

from the 2 paragraphs above, we can see now that we can assemble an inequality in x

24 < x < 30

If we look at the choices, we can immediately ignore 33 because x must be less than 30,

working out the choices, we find that the only choice which falls into the range 24<x<30 is the 2nd choice 12√5 (= 26.83) (which is the answer)

The last 2 choices give values smaller than 24 and are hence cannot be the answer