If the focus and directrix of a parabola are | vedclass

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(C) The focus is $S(3, 5)$ and the directrix is $x + y - 4 = 0$.The axis of the parabola passes through the focus and is perpendicular to the directri

Vedclass · Accepted Answer

$(2, 4)$

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(C) The focus is $S(3, 5)$ and the directrix is $x + y - 4 = 0$.The axis of the parabola passes through the focus and is perpendicular to the directrix. Since the directrix has a slope of $-1$,the axis has a slope of $1$.The equation of the axis is $y - 5 = 1(x - 3)$,which simplifies to $x - y + 2 = 0$.The foot of the directrix,$Z$,is the intersection of the axis and the directrix:$x - y + 2 = 0$$x + y - 4 = 0$Adding these equations gives $2x - 2 = 0$,so $x = 1$. Substituting $x = 1$ into $x + y = 4$ gives $y = 3$. Thus,$Z = (1, 3)$.The vertex $V$ is the midpoint of the segment $SZ$,where $S(3, 5)$ and $Z(1, 3)$.$V = \left( \frac{3 + 1}{2}, \frac{5 + 3}{2} \right) = (2, 4)$.

If the focus and directrix of a parabola are $(3, 5)$ and $x + y = 4$,then the coordinates of its vertex are

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