How to find an equation of a parabola given focus (10,1) and directrix x=5?

Question

Saniya Underwood · Accepted Answer

We use the following Focus-Directrix Property (FDP) of Parabola to solve this Problem :
FDP : Let a pt.

S

and a line

d,

be the focus & directrix of a parabola, resp. If

P

is any pt. on the parabola, then,

P

is equidistant from

S

and

d .

Focus is

S = S (10, 1) and the eqn. of dir. is d : x - 5 = 0 .

Let

P (x, y)

be any pt. on the parabola.

∴ the Dist. SP= \sqrt{{(x - 10)}^{2} + {(y - 1)}^{2}} ... ... ... ... ... ... ... . (1) .

The

⊥ - d i s t . from P to d is = \frac{| x - 5 |}{\sqrt{1^{2} + 0^{2}}} = | x - 5 | ... (2)

FDP

\Rightarrow \sqrt{{(x - 10)}^{2} + {(y - 1)}^{2}} = | x - 5 | .

\Rightarrow {(y - 1)}^{2} = {(x - 5)}^{2} - {(x - 10)}^{2} = 5 (2 x - 15), or,

y^{2} - 2 y - 10 x + 76 = 0 .

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