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If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x + y = a$$, is 16, then $$|a|$$ is equal to
The tangent at the vertex is $$x + y = a$$, which has slope $$-1$$. Since the axis of the parabola is perpendicular to this tangent, its slope is $$1$$.
Because the axis passes through the focus $$(a, a)$$ with slope 1, its equation is
$$y - a = 1 \cdot (x - a) \implies y = x$$The vertex lies on both the axis $$y = x$$ and the tangent at the vertex $$x + y = a$$, so
$$x + x = a \implies x = \frac{a}{2}$$Hence the vertex is $$\left(\dfrac{a}{2}, \dfrac{a}{2}\right)$$.
The distance between the vertex and the focus is
$$VF = \sqrt{\left(a - \frac{a}{2}\right)^2 + \left(a - \frac{a}{2}\right)^2} = \sqrt{\frac{a^2}{4} + \frac{a^2}{4}} = \frac{|a|}{\sqrt{2}}$$The length of the latus rectum is $$4 \times VF$$, so
$$4 \times \frac{|a|}{\sqrt{2}} = 16$$ $$\frac{|a|}{\sqrt{2}} = 4$$ $$|a| = 4\sqrt{2}$$Therefore, $$\boxed{|a| = 4\sqrt{2}}$$. The answer is Option C.
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