Question

A parabola has a focus of F(2,8.5) and a directrix of y=9.5. The point P(x,y) represents any point on the parabola, while D(x,9.5) represents any point on the directrix.

Maria was asked to use the distance formula to write an equation to represent this parabola.

Here is her work:

Step 1: FP=DP
Step 2: (x−2)2+(y−8.5)2−−−−−−−−−−−−−−−−√=(x−x)2+(y−9.5)2−−−−−−−−−−−−−−−−−√
Step 3: x2−4x+4+y2−17y+72.25=y2−19y+90.25
Step 4: x2−4x−14=−4y
Step 5: −14x2+x+72=y

Identify each incorrect step.

Select all answer choices that both state an incorrect step and explain why it is incorrect.

If there is only one incorrect step, select "only" and the answer choice that states and explains the incorrect step.


1,Step 3 is incorrect because she expanded the first binomial incorrectly.
2,only
3,Step 2 is incorrect because she used the wrong signs for the coordinates of the focus.
4,Step 4 is incorrect because she added the y-terms incorrectly.
5,Step 5 is incorrect because she divided incorrectly.
6,Step 3 is incorrect because a previous step is incorrect.
7,Step 5 is incorrect because a previous step is incorrect.
8, Step 4 is incorrect because a previous step is incorrect.

Answer 1

(4)and (7)

Explanation:

A parabola has a focus of F(2,8.5) and a directrix of y=9.5. The point P(x,y) represents any point on the parabola, while D(x,9.5) represents any point on the directrix.

Now, Step 1: FP=DP, using the distance formula,

Step 2:
`\sqrt{(y-8.5)^(2)+(x-2)^(2)}=\sqrt{(y-9.5)^(2)+(x-x)^(2)}`

Step 3: Solving the step 2,we get

`\sqrt{(y-8.5)^(2)+(x-2)^(2)}=\sqrt{(y-9.5)^(2) }`

Squaring on both sides and then solving, we get

`x^(2)-4x+4+y^(2)-17y+72.25=y^(2)-19y+90.25`

Step 4:
`x^(2)-4x-14=-2y`

Step 5: Finding the value of y by dividing both sides by -2, we get

`(x^(2)-4x-14)/(-2)=y`

Thus, Step 4 and 5 is incorrect because she added the y-terms incorrectly and also the previous step is incorrect on step 4.