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A parabola can be drawn given a focus of (-5, -4)(−5,−4) and a directrix of y=-6y=−6. Write the equation of the parabola in any form.
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A Parabola Can Be Drawn Given A Focus Of 5 454 And A Directrix Of Y6y6 Write The Equation Of The Parabola In Any Form Screenshot Below class=

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VectorFundament120

Answer:

[tex]\displaystyle \large{y=\dfrac{x^2}{4} + \dfrac{5x}{2} + \dfrac{5}{4}}[/tex]

Step-by-step explanation:

Given:

  • Focus = (-5,-4)
  • Directrix = -6

To find:

  • Parabola Equation

Locus of Parabola (Upward/Downward)

[tex]\displaystyle \large{\sqrt{(x-a)^2+(y-b)^2} = |y-c|}[/tex]

Where:

  • (a,b) = focus
  • c = directrix

Hence:

[tex]\displaystyle \large{\sqrt{(x+5)^2+(y+4)^2}=|y+6|}[/tex]

Cancel square root by squaring both sides as we get:

[tex]\displaystyle \large{(x+5)^2+(y+4)^2=(y+6)^2}[/tex]

Solve for y-term:

[tex]\displaystyle \large{(x+5)^2=(y+6)^2-(y+4)^2}\\\displaystyle \large{x^2+10x+25=y^2+12y+36-y^2-8y-16}\\\displaystyle \large{x^2+10x+25=4y+20}\\\displaystyle \large{x^2+10x+5=4y}\\\displaystyle \large{y=\dfrac{x^2}{4} + \dfrac{5x}{2} + \dfrac{5}{4}}[/tex]

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