Question

Question #4b5ea

Answer 1

`x^2 - 2xy + y^2 + 2 x + 2 y - 1 = 0`

Explanation:

Given:

focus: `(0,0)`
directrix: `x + y = 1`

Rewrite the equation for the directrix as:

`x+y-1=0`

This is done to that it fits the formula for the distance from a point to a line:

`d = |ax+by+c|/sqrt(a^2+b^2)" [1]"`

where `(x,y)` is a point on the parabola `a = 1, b = 1, and c = -1`

`d = |x+y-1|/sqrt(1^2+1^2)" [1.1]"`

`d = |x+y-1|/sqrt2" [1.2]"`

The distance from the focus, `(0,0)`, to the a point `(x,y)` on the parabola is:

`d = sqrt((x-0)^2+(y-0)^2)" [2]"`

`d = sqrt(x^2+y^2)" [2.1]"`

A parabola is the locus of points that are equidistant from its focus and its directrix, therefore, we can derive the equation of the parabola by setting the right side of equation [1.2] equal to the right side of equation [2.1]:

`|x+y-1|/sqrt2 = sqrt(x^2+y^2)`

Multiply both sides by `sqrt2`

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

Square both sides:

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

Expand the square:

`x^2 + 2 x y - 2 x + y^2 - 2 y + 1 = 2x^2+2y^2`

Combine like terms and write the equation in the General Cartesian form of a conic section , `Ax^2+Bxy + Cy^2+Dx +Ey + F= 0`:

`x^2 - 2xy + y^2 + 2 x + 2 y - 1 = 0`

Here is a graph of the parabola, the focus and the directrix: