Question

Any point on the parabola can be labeled (x,y) as shown.

1. What are the distances from the point (x,y)(x,y) to the focus of the parabola and the directrix?
Select two answers.


distance to the focus: √(x+3)2+(y−3)2

distance to the directrix: |y−4|

distance to the directrix: |x−4|

distance to the focus: √(x−2)2+(y+3)2

distance to the focus: √(x+3)2+(y−2)2

distance to the directrix: |y+4|

2. What is the correct standard form of the equation of the parabola?

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Answer 1

Heyo! :D

Distance to the directrix is |y - 4|.

Distance to the focus is
`√((x+3)^2+(y-2)^2)`.

Hope this helps! If so, please lmk! Tysm and good luck!

Answer 2

Part A)

2nd and 5th choice.

Part B)

`\displaystyle y=-(1)/(4)x^2-(3)/(2)x+(3)/(4)`

Explanation:

For the given parabola, our focus is (-3, 2), and our directrix is given by y = 4.

Part A)

We are given a point (x, y) on the parabola.

Since our directrix is an equation of y, the distance from (x, y) to the directrix will simply be the absolute value of the difference in y-values. So:

`d_1=|y-4|=|4-y|`

Recall the distance formula:

`d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2`

Our distance from (x, y) to the focus (-3, 2) can be determined using the distance formula. Let (x, y) be (x₂, y₂) and let our focus (-3, 2) be (x₁, y₁). Therefore:

`d_2=√((x-(-3))^2+(y-2)^2)=\sqrt{(x+3)^2+(y-2)^2`

Hence, for Part A, our answers are the 2nd and 5th choices.

Part B)

Recall that by the definition of a parabola, any point (x, y) on it is equidistant to the directrix and focus. Hence:

`√((x+3)^2+(y-2)^2)=|y-4|`

Solve for y. Square both sides. We may remove the absolute value since anything squared is positive:

`(x+3)^2+(y-2)^2=(y-4)^2`

Square:

`x^2+6x+9+y^2-4y+4=y^2-8y+16`

Rearrange:

`(x^2+6x+13)+(-16)=(y^2-y^2)+(-8y+4y)`

Combine like terms:

`x^2+6x-3=-4y`

Divide both sides by -4. Hence, our equation is:

`\displaystyle y=-(1)/(4)x^2-(3)/(2)x+(3)/(4)`