The length of the latus rectum of a parabola | vedclass

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(B) The distance $d$ between the focus $(x_1, y_1)$ and the directrix $ax + by + c = 0$ is given by $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$.H

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$ 3\sqrt{2} $

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(B) The distance $d$ between the focus $(x_1, y_1)$ and the directrix $ax + by + c = 0$ is given by $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$.Here,the focus is $(3, -4)$ and the directrix is $x + y - 2 = 0$.So,$d = \frac{|1(3) + 1(-4) - 2|}{\sqrt{1^2 + 1^2}} = \frac{|3 - 4 - 2|}{\sqrt{2}} = \frac{|-3|}{\sqrt{2}} = \frac{3}{\sqrt{2}}$.The length of the latus rectum of a parabola is $4$ times the distance from the focus to the directrix,but in standard geometry,the distance from the focus to the vertex is $a$,and the distance from the focus to the directrix is $2a$. The length of the latus rectum is $4a$.Therefore,the length of the latus rectum $= 2 	imes (	ext{distance between focus and directrix}) = 2 	imes \frac{3}{\sqrt{2}} = 3\sqrt{2}$.

The length of the latus rectum of a parabola whose directrix is $x + y - 2 = 0$ and focus is $(3, -4)$ is:

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