Let (1, 2) be the focus and x+y+1=0 be the di | vedclass

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(D) Let $P(x, y)$ be any point on the hyperbola $H$. The focus is $S(1, 2)$ and the directrix is $x+y+1=0$. By the definition of a conic section,$\fra

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$x^2+6xy+y^2+10x+14y-7=0$

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(D) Let $P(x, y)$ be any point on the hyperbola $H$. The focus is $S(1, 2)$ and the directrix is $x+y+1=0$. By the definition of a conic section,$\frac{PS}{PM} = e$,where $e = \sqrt{3}$ is the eccentricity and $PM$ is the perpendicular distance from $P$ to the directrix. Thus,$PS^2 = e^2 PM^2$. $(x-1)^2 + (y-2)^2 = 3 \left( \frac{x+y+1}{\sqrt{1^2+1^2}} \right)^2$. $(x^2-2x+1) + (y^2-4y+4) = 3 \left( \frac{(x+y+1)^2}{2} \right)$. $2(x^2+y^2-2x-4y+5) = 3(x^2+y^2+1+2xy+2x+2y)$. $2x^2+2y^2-4x-8y+10 = 3x^2+3y^2+6xy+6x+6y+3$. Rearranging the terms,we get $x^2+6xy+y^2+10x+14y-7=0$.

Let $(1, 2)$ be the focus and $x+y+1=0$ be the directrix of a hyperbola $H$. If $\sqrt{3}$ is the eccentricity of $H$,then its equation is

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