Question

Derive the equation of the parabola with a focus at (2, −1) and a directrix of y = − one half . (2 points)



f(x) = −(x + 2)2 − three fourths

f(x) = (x + 2)2 + three fourths

f(x) = −(x − 2)2 + three fourths

f(x) = −(x − 2)2 − three fourths

Answer 1

Focus of the parabola is (2,-1) and directrix is y = -1/2.

Let's assume a point (x,y) on parabola.

According to definition of parabola, the distance between point (x,y) and focus (2,-1) would be same as the distance between the point (x,y) and directrix y = -1/2.

$\sqrt{(x-2)^2+(y+1)^2} = \sqrt{(y+\frac{1}{2}) ^2} \\\\(x-2)^2+(y+1)^2=(y+\frac{1}{2} )^2 \\\\(x-2)^2 + y^2 + 2y +1 = y^2 +y +\frac{1}{4} \\\\(x-2)^2 =  -y^2 - 2y -1 + y^2 +y +\frac{1}{4} \\\\(x-2)^2 =  -y -1 +\frac{1}{4} \\\\(x-2)^2 =  -y +\frac{-4+1}{4} \\\\(x-2)^2 =  -y +\frac{-3}{4} \\\\y =  -(x-2)^2  +\frac{-3}{4}$

Hence, option D is correct, i.e. f(x) = −(x − 2)² − three fourths.

Answer 2

Answer:

$y = - (x - 2)^2 - \frac{3}{4}$

Step-by-step explanation:

The parabola must have all the points to be equally distant from the point of focus and the directrix, where directrix a horizontal line and the focus being a point given.

To derive an equation for the parabola, we will use the formula of distance. Now we have a random point on the parabola (x,y) and the point (x,y) will be equidistant from the focus and the directrix. So using the distance formula, we will get:

$\sqrt{(y -(\frac{-1}{2}}))^2} = \sqrt{(x-2)^2 + (y - (-1)^2}$

The square root of y-(-1/2) from the directrix, and the righthand side of the equal sign is derived from the focus point. Now simplify it to get:

$(y + \frac{1}{2} )^2 = (x - 2)^2 + (y + 1)^2$

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

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

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

$y = - (x - 2)^2 - \frac{3}{4}$

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