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(C) The given equation of the parabola is $y^2 - 4y - 3x + 7 = 0$.Rewriting it as $(y - 2)^2 = 3x - 3 = 3(x - 1)$,which is of the form $(y - k)^2 = 4a

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$\frac{1}{5}$

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(C) The given equation of the parabola is $y^2 - 4y - 3x + 7 = 0$.Rewriting it as $(y - 2)^2 = 3x - 3 = 3(x - 1)$,which is of the form $(y - k)^2 = 4a(x - h)$.Here,$4a = 3$,so $a = \frac{3}{4}$. The vertex is $(h, k) = (1, 2)$.The focus is $(h + a, k) = (1 + \frac{3}{4}, 2) = (\frac{7}{4}, 2)$.The focal chord passes through the focus $(\frac{7}{4}, 2)$ and the point $(4, 5)$.The slope of the chord is $m = \frac{5 - 2}{4 - 7/4} = \frac{3}{9/4} = \frac{12}{9} = \frac{4}{3}$.The equation of the chord is $y - 5 = \frac{4}{3}(x - 4)$,which simplifies to $3y - 15 = 4x - 16$,or $4x - 3y - 1 = 0$.The perpendicular distance from the origin $(0, 0)$ to the line $4x - 3y - 1 = 0$ is given by $d = \frac{|4(0) - 3(0) - 1|}{\sqrt{4^2 + (-3)^2}} = \frac{|-1|}{\sqrt{16 + 9}} = \frac{1}{\sqrt{25}} = \frac{1}{5}$.

The perpendicular distance from the origin to the focal chord drawn through the point $(4, 5)$ to the parabola $y^2 - 4y - 3x + 7 = 0$ is

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