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:
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]
