Question

WHO LIKES PARABOLAS 3 MULTIPLE CHOICE

1. What is the vertex of the parabola? y−3=12(x+5)^2
2. The equation of a parabola is y=12x2+6x+23 .
What is the equation of the directrix?
y = 5.5
y = 4.5
​y = 2​
​y = 0.5
3.The equation of a parabola is 132(y−2)2=x−1 .
What are the coordinates of the focus?
(−7, 2)
(1, 10)
(1, −6)
(9, 2)

Answer 1

y=4.5

I took the test

Answer 2

Question 1) Vertex of the Parabola.


`y-3=12(x+5)^(2) \\ \\ y= 12(x+5)^(2)+3`

The vertex of the general parabolic equation:


`y=a(x-h)^(2)+k`

lies at (h,k)

Comparing the given equation to general equation, we can write:

h = - 5
k = 3

So, the vertex of the given parabola will be (-5, 3)

Question 2) Equation of Directrix

The correct equation of the parabola is:


`y= (1)/(2) x^(2) +6x+23 \\ \\`

First we need to convert the equation to standard form as shown below:


`y= (1)/(2)( x^(2) +12x)+23 \\ \\ y= (1)/(2)( x^(2) +12x+36)+23- (1)/(2)(36) \\ \\ y= (1)/(2)(x+6)^(2)+5 \\ \\ y-5= (1)/(2)(x+6)^(2) \\ \\ 2(y-5)=(x+6)^(2) \\ \\ 4( (1)/(2))(y-5)= (x+6)^(2)`

The directrix of the general parabola of the form:


`4p(y-k)=(x-h)^(2)`

is y = k - p

Comparing equation of given parabola with general parabolic equation, we can write:
p =1/2
k = 5

So, equation of directirx will be:

y = 5 - 1/2 = 4.5

So, option B gives the correct answer.

Question 3) Focus of the parabola

The correct equation of the parabola is:


`(1)/(32)(y-2)^(2)=x-1 \\ \\ (y-2)^(2)=32(x-1) \\ \\ (y-2)^(2)=4*8(x-1)`

Comparing this equation to the general parabolic equation, we can write:
p=8
h =1
k = 2

The focus of the parabola will be at (h+p,k) = (9,2)

So the focus of the parabola is at (9,2)

Thus, option D gives the correct answer.