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Which of the following is the equation of the parabola described with vertex at $(5, -3)$ , axis parallel to the $y$ -axis and passing through the point $(1, 1)$ ? $\newline$ (a) $(x - 5)^2 = 4(y + 3)$ $\newline$ (b) $(x + 5)^2 = 4(y - 3)$ $\newline$ (c) $(y + 3)^2 = 4(x - 5)$ $\newline$ (d) $(y - 3)^2 = 4(x + 5)$

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Convert equations of conic sections from general to standard form

Full solution

Q. Which of the following is the equation of the parabola described with vertex at

(5, -3)

, axis parallel to the

y

-axis and passing through the point

(1, 1)

?

\newline

(a)

(x - 5)^2 = 4(y + 3)

\newline

(b)

(x + 5)^2 = 4(y - 3)

\newline

(c)

(y + 3)^2 = 4(x - 5)

\newline

(d)

(y - 3)^2 = 4(x + 5)

Identify standard form: Step $1$ : Identify the standard form of the equation for a parabola with a vertical axis. The standard form is $(x - h)^2 = 4p(y - k)$ , where $(h, k)$ is the vertex.
Plug in vertex: Step $2$ : Plug in the vertex $(5, -3)$ into the equation. This gives us $(x - 5)^2 = 4p(y + 3)$ .
Find value of $p$ : Step $3$ : Use the point $(1, 1)$ to find the value of $p$ . Substitute $x = 1$ and $y = 1$ into the equation: $(1 - 5)^2 = 4p(1 + 3)$ .
Simplify and solve: Step $4$ : Simplify and solve for $p$ . $(1 - 5)^2 = 16$ , and $1 + 3 = 4$ , so $16 = 4p \times 4$ .
Solve for $p$ : Step $5$ : Solve $16 = 16p$ . Divide both sides by $16$ , $p = 1$ .
Substitute $p$ back: Step $6$ : Substitute $p$ back into the equation. We get $(x - 5)^2 = 4(y + 3)$ .

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