What is the equation in standard form of the parabola with a focus at (2,3) and a directrix of y= 9?

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What is the equation in standard form of the parabola with a focus at (2,3) and a directrix of y= 9?

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Answer 1 · 2017-08-30T16:13:39

$x^{2} - 4 x + 12 y - 68 = 0$ $x^2-4x+12y-68=0$

Explanation:

$for any point (x, y) on the parabola$ $"for any point "(x,y)" on the parabola"$

$the distance from (x . y) to the focus and directrix$ $"the distance from "(x.y)" to the focus and directrix"$
$are equal$ $"are equal"$

$using the distance formula$ $"using the "color(blue)"distance formula"$

$with (x, y) \to (2, 3)$ $"with "(x,y)to(2,3)$

$\Rightarrow \sqrt{{(x - 2)}^{2} + {(y - 3)}^{2}} = | y - 9 |$ $rArrsqrt((x-2)^2+(y-3)^2)=|y-9|$

$squaring both sides$ $color(blue)"squaring both sides"$

${(x - 2)}^{2} + {(y - 3)}^{2} = {(y - 9)}^{2}$ $(x-2)^2+(y-3)^2=(y-9)^2$

$\Rightarrow x^{2} - 4 x + 4 + y^{2} - 6 y + 9 = y^{2} - 18 y + 81$ $rArrx^2-4x+4+y^2-6y+9=y^2-18y+81$

$\Rightarrow x^{2} - 4 x + 12 y - 68 = 0$ $rArrx^2-4x+12y-68=0$

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What is the equation in standard form of the parabola with a focus at (2,3) and a directrix of y= 9?

1 Answer

Explanation:

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