Question

PLEASE HELP!!

Answer 1

Answer:

Part a) the equation of parabola is: $(y-3)^2 = 4(x-3)$

Part b) the equation of parabola is: $(y+1)^2 = -12 (x-5)$

Step-by-step explanation:

Part a: focus at (4, 3), and directrix x = 2

Step 1: Find the horizontal or vertical direction

As the directrix x=2 is horizontal, we will use the equation of horizontal parabola i.e.

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

Step 2: Find the vertex (h,k)

The vertex (h,k) is between the directrix and the focus

x-coordinate (h) can be found by:

$h=\frac{x-coordinate of focus + directrix}{2}\\h= \frac{4+2}{2}\\h=\frac{6}{2}\\h=3$

the value y-coordinate (k) will be the same as the y-coordinate of focus i.e. k=3

so, vertex(h,k) =(3,3)

Step 3: Find the distance from focus to vertex

The distance from focus to vertex is denoted by p

and can be found by:

p= x-coordinate of focus-x-coordinate of vertex

p= 4 - 3

p= 1

Step 4: Putting values of vertex (h,k) and p in the equation.

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

$(y-3)^2 = 4(1) (x-3)$

$(y-3)^2 = 4 (x-3)$

so, the equation of parabola is: $(y-3)^2 = 4(x-3)$

Part b: focus at (2, -1), and directrix x = 8

Step 1: Find the horizontal or vertical direction

As the directrix x=8 is horizontal, we will use the equation of horizontal parabola i.e.

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

Step 2: Find the vertex (h,k)

The vertex (h,k) is between the directrix and the focus

x-coordinate (h) can be found by:

$h=\frac{x-coordinate of focus + directrix}{2}\\h= \frac{2+8}{2}\\h=\frac{10}{2}\\h=5$

the value y-coordinate (k) will be the same as the y-coordinate of focus i.e. k=-1

so, vertex(h,k) =(5,-1)

Step 3: Find the distance from focus to vertex

The distance from focus to vertex is denoted by p

and can be found by:

p= x-coordinate of focus-x-coordinate of vertex

p= 2 - 5

p= -3

Step 4: Putting values of vertex (h,k) and p in the equation.

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

$(y-(-1))^2 = 4(-3) (x-5)$

$(y+1)^2 = -12 (x-5)$

So the equation of parabola is: $(y+1)^2 = -12 (x-5)$