If (1,2) is the focus,x+2y=0 is the directrix | vedclass

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(B) The definition of a conic section is $SP^2 = e^2 PM^2$,where $S$ is the focus,$P(x,y)$ is a point on the curve,$e$ is the eccentricity,and $PM$ is

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$3x^2-8xy-3y^2-10x-20y+25=0$

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(B) The definition of a conic section is $SP^2 = e^2 PM^2$,where $S$ is the focus,$P(x,y)$ is a point on the curve,$e$ is the eccentricity,and $PM$ is the perpendicular distance from $P$ to the directrix.Given focus $S = (1,2)$,directrix $x+2y=0$,and $e = \sqrt{2}$.$(x-1)^2 + (y-2)^2 = (\sqrt{2})^2 \frac{(x+2y)^2}{1^2+2^2}$$(x^2-2x+1) + (y^2-4y+4) = 2 \frac{(x+2y)^2}{5}$$5(x^2+y^2-2x-4y+5) = 2(x^2+4y^2+4xy)$$5x^2+5y^2-10x-20y+25 = 2x^2+8y^2+8xy$$3x^2-8xy-3y^2-10x-20y+25 = 0$

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

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