What is the standard form of the equation of the parabola with a directrix at x=6 and a focus at (9,5)?

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Answer 1 · 2017-06-28T13:53:09

$6x=y^2-10y+70$

Parabola is the locus of a point which moves so that its distance from a given point called focus and a given line called directrix is always same.

Let the point on parabola be $(x,y)$ . Here focus is $(9,5)$ and its distance from focus is $sqrt((x-9)^2+(y-5)^2)$ .

And as directrix is $x=6$ and distance of $(x,y)$ from $x=6$ is $|x-6|$ . Hence equation of parabola is

$(x-9)^2+(y-5)^2=(x-6)^2$

or $x^2-18x+81+y^2-10y+25=x^2-12x+36$

or $cancelx^2+81+y^2-10y+25-36=cancelx^2+18x-12x$

or $6x=y^2-10y+70$

or $x=1/6(y^2-10y+70)$

graph{(y^2-10y+70-6x)(x-6)((x-9)^2+(y-5)^2-0.03)=0 [-0.83, 19.17, -0.36, 9.64]}