Question

What is the equation of the quadratic graph with a focus of (4,3) and a directrix of y=-6

Answer 1

The correct option is B)
`y=(x^(2))/(6)-(4x)/(3)-(11)/(6)`

Explanation:

We need to find out the quadratic graph with given focus
`(4,-3)` and a directrix
`y=-6`

Using the distance formula, we find that the distance between (x,y) and the focus (4,-3) is
`\sqrt{(x-4)^(2)+(y+3)^(2)}` and the distance between (x,y) and the directrix y=-6, is
`\sqrt{(y+6)^(2)}`

On the parabola, these distances are equal:

`\sqrt{(y+6)^(2)}=\sqrt{(x-4)^(2)+(y+3)^(2)}`

`(y+6)^(2)=(x-4)^(2)+(y+3)^(2)`

`y^(2)+36+12y=x^(2)+16-8x+y^(2)+9+6y`

Simplify the above

`36+12y-6y=x^(2)-8x+9+16`

`6y=x^(2)-8x+9+16-36`

`6y=x^(2)-8x-11`

Divide both the sides by 6,

`y=(x^(2))/(6)-(8x)/(6)-(11)/(6)`

Simplified further,

`y=(x^(2))/(6)-(4x)/(3)-(11)/(6)`

Therefore, the correct option is B)
`y=(x^(2))/(6)-(4x)/(3)-(11)/(6)`