Question

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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Answer 1

`\displaystyle \large{y=(x^2)/(4) + (5x)/(2) + (5)/(4)}`

Explanation:

Given:

• Focus = (-5,-4)
• Directrix = -6

To find:

• Parabola Equation

Locus of Parabola (Upward/Downward)

`\displaystyle \largey-c`

Where:

• (a,b) = focus
• c = directrix

Hence:

`\displaystyle \large√((x+5)^2+(y+4)^2)=`

Cancel square root by squaring both sides as we get:

`\displaystyle \large{(x+5)^2+(y+4)^2=(y+6)^2}`

Solve for y-term:

`\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=(x^2)/(4) + (5x)/(2) + (5)/(4)}`