Question

What is the equation of the parabola with a focus at (2,1) and a directrix of y= 3?

Answer 1

`x^2-4x+4y-4=0`

Explanation:

`"for any point "(x,y)" on the parabola"`

`"the distance from "(x,y)" to the focus and directrix are"`
`"equal"`

`"using the "color(blue)"distance formula"`

`rArrsqrt((x-2)^2+(y-1)^2)=|y-3|`

`color(blue)"squaring both sides"`

`(x-2)^2+(y-1)^2=(y-3)^2`

`rArrx^2-4x+4+y^2-2y+1=y^2-6y+9`

`rArrx^2-4xcancel(+y^2)cancel(-y^2)-2y+6y+4+1-9=0`

`rArrx^2-4x+4y-4=0larrcolor(red)" is the equation"`