Question

Question #45399

Answer 1

Because the vertex is the origin and the rotation is centered about the origin, the vertex of the rotated parabola is the origin. The explanation gives much greater detail.

Explanation:

Please read this reference regarding the Rotation of Axes of conic sections.

The given equation

`x=y^2`

Written in the General Cartesian form

#Ax^2+Bxy+Cy^2+Dx+Ey + F = 0

Is:

`y^2 -x = 0`

Please observe that `A = B=E=F=0`, C = 1, and `D=-1`

Use equations 9.4.4a through 9.4.4f to compute the new values, A' through F' with `theta = 45^@`:

`A' = (A + C)/2 + [(A - C)/2] cos(2θ) - B/2 sin(2θ) = 1/2`
`B' = (A - C) sin(2θ) + B cos(2θ) = -1`
`C' = (A + C)/2 + [(C - A)/2] cos(2θ) + B/2 sin(2θ)= 1/2`
`D' = D cos(θ) - E sin(θ) = -sqrt2/2`
`E' = D sin(θ) + E cos(θ) = -sqrt2/2`
`F'=F = 0`

The rotated parabola is:

`1/2x^2-xy+1/2y^2-sqrt2/2x-sqrt2/2y = 0`

Return to the original parabola `x = ay^2` where `a = 1`

The focal distance is `f = 1/(4a) = 1/4`
This makes the focus of the original parabola `(h+f,k)=(1/4,0)`
The directrix of the original parabola is `x = h-f = -1/4`

The new focus will be `(1/4cos(45^@), 1/4sin(45^@)) = (sqrt2/8,sqrt2/8)`

The angle that the original directrix forms with the x axis is `90^@`; this means that the new directrix must form an angle of `135^@`

`y = tan(135^@)x+b`

`y +x - b = 0`

Use the distance from a point to a line:

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

Where the distance is `2f= 1/2` from the focus `(sqrt2/8,sqrt2/8)`:

`1/2 = |sqrt2/8+sqrt2/8-b|/sqrt(1^2+1^2)`

`sqrt2/2 = |sqrt2/4-b|`

`b = +-sqrt2/4`

The positive value does not make sense:

`b = -sqrt2/4`

`x+y=-sqrt2/4`

Here is the original equation with is focus and directrix:

Here is the rotated equation with its focus and directrix:

Please observe that both of them have their vertex at the origin.