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(C) The slope of the directrix $x+y+1=0$ is $-1$. Since the axis of a parabola is perpendicular to the directrix,the slope of the axis is $1$.Given th

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$4$

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(C) The slope of the directrix $x+y+1=0$ is $-1$. Since the axis of a parabola is perpendicular to the directrix,the slope of the axis is $1$.Given the vertex is $(1, 1)$,the equation of the axis is $y-1=1(x-1)$,which simplifies to $y=x$.To find the point $(c, d)$,we solve the system of equations for the directrix and the axis:$x+y+1=0$ and $y=x$.Substituting $y=x$ into the directrix equation: $x+x+1=0$ $\Rightarrow 2x=-1$ $\Rightarrow x=-\frac{1}{2}$.Thus,$c=-\frac{1}{2}$ and $d=-\frac{1}{2}$.The vertex $(1, 1)$ is the midpoint of the focus $(a, b)$ and the point $(c, d)$.Using the midpoint formula: $1=\frac{a+c}{2}$ $\Rightarrow 1=\frac{a-1/2}{2}$ $\Rightarrow 2=a-\frac{1}{2}$ $\Rightarrow a=\frac{5}{2}$.Similarly,$1=\frac{b+d}{2}$ $\Rightarrow 1=\frac{b-1/2}{2}$ $\Rightarrow 2=b-\frac{1}{2}$ $\Rightarrow b=\frac{5}{2}$.Finally,$a+b+c+d = \frac{5}{2} + \frac{5}{2} - \frac{1}{2} - \frac{1}{2} = 5 - 1 = 4$.

$(1, 1)$ is the vertex and $x+y+1=0$ is the directrix of a parabola. If $(a, b)$ is its focus and $(c, d)$ is the point of intersection of the directrix and the axis of the parabola,then $a+b+c+d=$

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