If y -4 is directrix and 0-2 be the vertex of parabola x 2- y -0- then the value of - is

V is the midpoint of AS (where S is the focus) ⇒ S(0,0) Let P(h,k) be any point on the parabola nbsp; PS=PM nbsp; nbsp;(∵directrix of the parabola) Wher ...

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Standard XII

Mathematics

Parametric Equation of Parabola

If y =4 is di...

Question

If $y = 4$ is directrix and $(0, 2)$ be the vertex of parabola $x^{2} + λ y + μ = 0$ , then the value of $λ - μ$ is

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Solution

V is the midpoint of $A S$ (where $S$ is the focus)
$\Rightarrow S (0, 0)$
Let $P (h, k)$ be any point on the parabola
$P S = P M$ ( $∵$ directrix of the parabola)
Where $M$ is foot of perpendicular from $P$ on directrix.
$\sqrt{(h - 0)^{2} + (k - 0)^{2}} = | k - 4 | \Rightarrow h^{2} + k^{2} = k^{2} - 8 k + 16 \Rightarrow x^{2} + 8 y - 16 = 0 λ = 8, μ = - 16$

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If y=4 is directrix and (0,2) be the vertex of parabola x2+λy+μ=0 , then the value of λ−μ is

V is the midpoint of AS (where S is the focus) ⇒S(0,0) Let P(h,k) be any point on the parabola PS=PM (∵directrix of the parabola) Where M is foot of perpendicular from P on directrix. √(h−0)2+(k−0)2=|k−4|⇒h2+k2=k2−8k+16⇒x2+8y−16=0λ=8,μ=−16

If $y = 4$ is directrix and $(0, 2)$ be the vertex of parabola $x^{2} + λ y + μ = 0$ , then the value of $λ - μ$ is