If the equation of the directrix of a parabol | vedclass

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(D) The distance between the directrix $(3x + 4y + 15 = 0)$ and the tangent at the vertex $(3x + 4y - 5 = 0)$ is equal to the distance $a$ from the ve

Vedclass · Accepted Answer

$16$

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(D) The distance between the directrix $(3x + 4y + 15 = 0)$ and the tangent at the vertex $(3x + 4y - 5 = 0)$ is equal to the distance $a$ from the vertex to the focus.The distance $a$ is given by the formula for the distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$,which is $a = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.Substituting the values: $a = \frac{|15 - (-5)|}{\sqrt{3^2 + 4^2}} = \frac{20}{\sqrt{9 + 16}} = \frac{20}{5} = 4$.The length of the latus rectum of a parabola is $4a$.Therefore,the length of the latus rectum $= 4 	imes 4 = 16$.

If the equation of the directrix of a parabola is $3x + 4y + 15 = 0$ and the equation of the tangent at the vertex is $3x + 4y - 5 = 0$,then what is the length of the latus rectum?

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