Question

Determine the focus coordinates, the axis of the parabola, the equation of the directrix and the latus rectum length for y^2 = -8x

Answer 1

$\textrm{The given equation is y² = - 8x}$

$\textrm{Here, the coefficient of x is negative.}$

$\textrm{Hence, the parabola opens towards the left}$

$\textrm{On comparing this equation with y² = 4ax , we get,}$

$\sf {4a = -8 }\\ :\implies \sf{a = 2}$

$\textsf \pink{∴ Coordinates of the focus = (-a,0) =} \sf \color{purple}{ \left( -2,0 \right)}$

$\textrm{Axis of the parabola is the x-axis i.e y = 0}$

$\textrm{Equation of directrix, x = a i.e. x = 2 }$

$: \boxed{ \textsf{\red{Length of latus rectum = 4a = 8}}}$

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Answer 2

The given equation is y² = - 8x

\textrm{Here, the coefficient of x is negative.}Here, the coefficient of x is negative.

\textrm{Hence, the parabola opens towards the left}Hence, the parabola opens towards the left

\textrm{On comparing this equation with y² = 4ax , we get,}On comparing this equation with y² = 4ax , we get,

\begin{gathered} \sf {4a = -8 }\\ :\implies \sf{a = 2} \\\end{gathered}

4a=−8

:⟹a=2

\begin{gathered} \textsf \pink{∴ Coordinates of the focus = (-a,0) =} \sf \color{purple}{ \left( -2,0 \right)} \\ \end{gathered}

∴ Coordinates of the focus = (-a,0) =(−2,0)

\textrm{Axis of the parabola is the x-axis i.e y = 0}Axis of the parabola is the x-axis i.e y = 0

\textrm{Equation of directrix, x = a i.e. x = 2 }Equation of directrix, x = a i.e. x = 2

: \boxed{ \textsf{\red{Length of latus rectum = 4a = 8}}}:

Length of latus rectum = 4a = 8

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