The point of intersection of the latus rectum | vedclass

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Question

(A) The point of intersection of the latus rectum and the axis of a parabola is its focus.Given the equation of the parabola: ${y^2} + 4x + 2y - 8 = 0

Vedclass · Accepted Answer

$(5/4, -1)$

Vedclass · Answer

(A) The point of intersection of the latus rectum and the axis of a parabola is its focus.Given the equation of the parabola: ${y^2} + 4x + 2y - 8 = 0$.Rearranging the terms to complete the square for $y$:${y^2} + 2y = -4x + 8$${y^2} + 2y + 1 = -4x + 8 + 1$${(y + 1)^2} = -4x + 9$${(y + 1)^2} = -4(x - 9/4)$.Comparing this with the standard form ${(y - k)^2} = 4a(x - h)$,we get $h = 9/4$,$k = -1$,and $4a = -4$,which implies $a = -1$.The focus of the parabola is given by $(h + a, k)$.Substituting the values: $(9/4 - 1, -1) = (5/4, -1)$.

The point of intersection of the latus rectum and the axis of the parabola ${y^2} + 4x + 2y - 8 = 0$ is:

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