What is the standard form of the equation of the parabola with a directrix at x=5 and a focus at (11,-7)?

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Answer 1 · 2016-05-22T23:01:28

${(y + 7)}^{2} = 12 \cdot (x - 8)$ $(y+7)^2=12*(x-8)$

Your equation is of the form

${(y - k)}^{2} = 4 \cdot p \cdot (x - h)$ $(y-k)^2=4*p*(x-h)$

The focus is $(h + p, k)$ $(h+p, k)$

The directrix is $(h - p)$ $(h-p)$

Given the focus at $(11, - 7) \to h + p = 11 and k = - 7$ $(11,-7) -> h+p= 11 " and " k=-7$

The directrix $x = 5 \to h - p = 5$ $x=5 -> h-p = 5$

$h + p = 11 (e q . 1)$ $h+p= 11" " (eq. 1)"$
$h - p = 5 (e q . 2)$ $h-p =5 " " (eq. 2)$

$ul("use (eq. 2) and solve for h")$

$" " h=5+p" (eq. 3)"$

$ul("Use (eq. 1) + (eq. 3) to find the value of " p)$

$(5+p)+p=11$

$5+2p=11$

$2p=6$

$p=3$

$ul("Use (eq.3) to find the value of " h )$

$h=5+p$

$h=5+3$

$h=8$

$"Plugging the values of " h, p " and " k " in the equation "(y-k)^2=4*p*(x-h) " gives"$

$(y-(-7))^2=4*3*(x-8)$

$(y+7)^2=12*(x-8)$