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A parabola is defined as the set of points in the plane that are equidistant from a fixed point (called the focus of the parabola) and a fixed line (called the directrix of the parabola).Consider the parabola with focus point (1,1) and directrix the horizontal line y=−3.Plot the focus and draw the directrix graph.

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isyllus

Answer:

[tex]y=\dfrac{1}{8}(x-1)^2-1[/tex]

Step-by-step explanation:

A parabola is defined as the set of points in the plane that are equidistant from a fixed point  and a fixed line.

  • Fixed point called focus.
  • Fixed line called directirx.

Let point on parabola be (x,y)

Distance from focus(1,1) and point (x,y):

[tex]d=\sqrt{(x-1)^2+(y-1)^2}[/tex]

Distance from point (x,y) and line y=-3  ( x would be vary) (x,-3)

[tex]d=\sqrt{(x-x)^2+(y+3)^2}[/tex]

Both distance must be equal for parabola

[tex]\sqrt{(x-1)^2+(y-1)^2}=\sqrt{(x-x)^2+(y+3)^2}[/tex]

[tex](x-1)^2+(y-1)^2=(y+3)^2[/tex]

[tex]x^2+1-2x+y^2+1-2y=y^2+9+6y[/tex]

[tex]x^2+2x+2=9+6y+2y[/tex]

[tex]y=\dfrac{1}{8}(x^2+2x-7)[/tex]

[tex]y=\dfrac{1}{8}(x-1)^2-1[/tex]

Please find attachment for graph. Focus and directrix shown in graph.

Ver imagen isyllus

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