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Question Number 112664 by bemath last updated on 09/Sep/20

$$\mathrm{what}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{parabola}$$$$\mathrm{whose}\mathrm{focus}\mathrm{F}(-3,4)\mathrm{and}\mathrm{directrix}$$$$\mathrm{is}3\mathrm{x}-4\mathrm{y}+5=0?$$

Commented by bemath last updated on 09/Sep/20

Answered by bobhans last updated on 09/Sep/20

$$\left(⧫\right)\sqrt{{(\mathrm{x}+3)}^2+{(\mathrm{y}-4)}^2}=\mid \frac{3\mathrm{x}-4\mathrm{y}+5}{5}\mid$$$$\mathrm{squaring}\Rightarrow {(\mathrm{x}-3)}^2+{(\mathrm{y}-4)}^2=\frac{{(3\mathrm{x}-4\mathrm{y}+5)}^2}{25}$$$$\Rightarrow 25({\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}-8\mathrm{y}+25)={9\mathrm{x}}^2+{16\mathrm{y}}^2+25-24\mathrm{xy}+30\mathrm{x}-40\mathrm{y}$$$$\Rightarrow {16\mathrm{x}}^2+{9\mathrm{y}}^2+24\mathrm{xy}-150\mathrm{x}-160\mathrm{y}+600=0$$

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