Question

What is the eqution of the parabola with focus (3,2) and directrix 3x-4y+9=0 ?

Answer 1

`16x^2 + 24x y + 9y^2 - 204x - 28y + 244 = 0`

Explanation:

We know that a parabola is the locus of points equidistant from its focus and its directrix.

The distance from the focus, `(3,2)`, to any point, `(x,y)`, on the parabola is:

`d = sqrt((x-3)^2+(y-2)^2)" [1]"`

The distance from the line, `3x - 4y+9 = 0` to any point `(x,y)` is:

`d = |3x-4y+9|/sqrt(3^2+(-4)^2)" [2]"`

Because `d = d`, we can set the right side of equation [1] equal to the right side of equation [2]:

`sqrt((x-3)^2+(y-2)^2)= |3x-4y+9|/sqrt(3^2+(-4)^2)`

An alternative form of the absolute value function is `|A| = sqrt(A^2)`:

`sqrt((x-3)^2+(y-2)^2)= sqrt((3x-4y+9)^2)/sqrt(3^2+(-4)^2)`

Squaring both sides of the equation:

`(x-3)^2+(y-2)^2= (3x-4y+9)^2/(3^2+(-4)^2)`

Simplify the denominator:

`(x-3)^2+(y-2)^2= (3x-4y+9)^2/25`

Multiply both sides by 25:

`25(x-3)^2+25(y-2)^2= (3x-4y+9)^2`

Expanding the squares and combining like terms:

`16x^2 + 24x y + 9y^2 - 204x - 28y + 244 = 0`

This is the standard Cartesian form for a conic section; it is a parabola with rotated axes:

Answer 2

Equation of parabola is `16x^2+9y^2+24xy-204x-28y+244=0`

Explanation:

Parabola is the locus of a point which movesso that its distance from a given pint called focus and a given line called directrix is always equal.

Here let the pont be `(x,y)` and its distance from focus at `(3,2)` is

`sqrt((x-3)^2+(y-2)^2)`

and its distance from `3x-4y+9=0` is

`|(3x-4y+9)/sqrt(3^2+4^2)|=|(3x-4y+9)/5|`

Hence, equation of parabola is

`(x-3)^2+(y-2)^2=(3x-4y+9)^2/25`

or `25(x^2-6x+9+y^2-4y+4)=9x^2+16y^2+81-24xy-72y+54x`

i.e. `16x^2+9y^2+24xy-204x-28y+244=0`

graph{((x-3)^2+(y-2)^2-0.03)(3x-4y+9)(16x^2+9y^2+24xy-204x-28y+244)=0 [-8.79, 11.21, -2.56, 7.44]}