Question

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

Answer 1

Answer:

Part A)

2nd and 5th choice.

Part B)

$\displaystyle y=-\frac{1}{4}x^2-\frac{3}{2}x+\frac{3}{4}$

Step-by-step 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:

$d_2=\sqrt{(x-(-3))^2+(y-2)^2}=\sqrt{(x+3)^2+(y-2)^2$

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:

$\sqrt{(x+3)^2+(y-2)^2}=|y-4|$

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:

$\displaystyle y=-\frac{1}{4}x^2-\frac{3}{2}x+\frac{3}{4}$