Find the parametric coordinates of any point | vedclass

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(D) The focus is $S(0, 1)$ and the directrix is $x + 2 = 0$.The definition of a parabola is $SP = PM$,where $P(x, y)$ is a point on the parabola.$SP^2

Vedclass · Accepted Answer

$(t^2 - 1, 2t + 1)$

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(D) The focus is $S(0, 1)$ and the directrix is $x + 2 = 0$.The definition of a parabola is $SP = PM$,where $P(x, y)$ is a point on the parabola.$SP^2 = PM^2 \implies (x - 0)^2 + (y - 1)^2 = (x + 2)^2$$x^2 + (y - 1)^2 = x^2 + 4x + 4$$(y - 1)^2 = 4(x + 1)$Let $Y = y - 1$ and $X = x + 1$. Then the equation becomes $Y^2 = 4X$,which is of the form $Y^2 = 4aX$ with $a = 1$.The parametric coordinates for $Y^2 = 4aX$ are $(at^2, 2at)$.Here,$X = t^2$ and $Y = 2t$.Substituting back for $x$ and $y$:$x + 1 = t^2 \implies x = t^2 - 1$$y - 1 = 2t \implies y = 2t + 1$Thus,the parametric coordinates are $(t^2 - 1, 2t + 1)$.

Find the parametric coordinates of any point on the parabola whose focus is $(0, 1)$ and directrix is $x + 2 = 0$.

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