What is the equation in standard form of the parabola with a focus at (42, -31) and a directrix of y=2?

Question

dicky23628u6a · Accepted Answer

Explanation:
Please observe that the directrix is a horizontal line
y=2
Therefore, the parabola is the type that opens upward or downward; the vertex form of the equation for this type is:

y = \frac{1}{4 f} {(x - h)}^{2} + k

[1]
Where (h,k) is the vertex and f is the signed vertical distance from the vertex to the focus.
The x coordinate of the vertex is the same as the x coordinate of the focus:
h=42
Substitute 42 for h into equation [1]:

y = \frac{1}{4 f} {(x - 42)}^{2} + k

[2]
The y coordinate of the vertex is halfway between the directrix and the focus:

k = \frac{y_{d i r e c t r i x} + y_{f o c u s}}{2}

k = \frac{2 + (- 31)}{2}

k = - \frac{29}{2}

Substitute

- \frac{29}{2}

for k into equation [2]:

y = \frac{1}{4 f} {(x - 42)}^{2} - \frac{29}{2}

[3]
The equation to find the value of f is:

f = y_{f o c u s} - k

f = - 31 - (- \frac{29}{2})

f = - \frac{33}{2}

Substitute

- \frac{33}{2}

for f into equation [3]:

y = \frac{1}{4 (- \frac{33}{2})} {(x - 42)}^{2} - \frac{29}{2}

Simplify the fraction:

y = - \frac{1}{66} {(x - 42)}^{2} - \frac{29}{2}

Expand the square:

y = - \frac{1}{66} (x^{2} - 84 x + 1764) - \frac{29}{2}

Distribute the fraction:

y = - \frac{1}{66} x^{2} + \frac{14}{11} x - \frac{294}{11} - \frac{29}{2}

Combine like terms:

y = - \frac{1}{66} x^{2} + \frac{14}{11} x - \frac{907}{22} \leftarrow

standard form

Heidy Prince · Accepted Answer

Explanation:
We will solve this Problem using the following Focus-Directrix
Property (FDP) of the Parabola.
FDP : Any point on a Parabola is equidistant from the
Focus and the Directrix.
Let, the point F=F(42, -31), and , the line d: y-2=0, be the Focus and the Directrix of the Parabola, say S.
Let,

P = P (x, y) \in

S, be any General Point.
Then, using the Distance Formula, we have, the distance,

F P = \sqrt{{(x - 42)}^{2} + {(y + 31)}^{2}}

...(1)
Knowing that the \bot - ist. between a point (k,k), and, a line :
ax+by+c=0, is,

\frac{| a h + b k + c |}{\sqrt{a^{2} + b^{2}}}

, we find that,
the

⊥

-dist. btwn P(x,y), &, d is, |y-2|...(2)
By FDP, (1), and (2), we have,

\sqrt{{(x - 42)}^{2} + {(y + 31)}^{2}} = | y - 2 |

, or,

{(x - 42)}^{2} = {(y - 2)}^{2} - {(y + 31)}^{2} = - 66 y - 957

, i.e.

x^{2} - 84 x + 1764 = - 66 y - 957

∴ 66 y = - x^{2} + 84 x - 2721

, which, in the Standard Form,
reads,

y = - \frac{1}{66} x^{2} + \frac{14}{11} x - \frac{907}{22}

What is the equation in standard form of the parabola

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