equation of tilted parabola

[galactus]

Here is a problem I recently ran across I thought was kind of interesting if anyone would like a go.

"find the equation of the parabola with directrix y=2x and focus at (5,2)".

 

[soroban]

Hello, galactus!

I'll give it a try . . .

$\displaystyle \text{Find the equation of the parabola with directrix }y\,=\,2x\,\text{ and focus at }(5,2).$

\(\displaystyle \text{We have the focus }F\!\!5,2)\:\text{ and the directrix }D\!\!:\,2x-y \:=\:0\)

$\displaystyle \text{The parabola is the set of all point }P(x,y)\,\text{ such that: }\,\overline{PF} \:=\:\overline{PD}.$

. . $\displaystyle \overline{PF} \:=\:\sqrt{(x-5)^2 + (y-2)^2}$

. . $\displaystyle \overline{PD} \:=\:\frac{2x-y}{\sqrt{5}}$


$\displaystyle \text{We have: }\qquad\; \sqrt{(x-5)^2 + (y-2)^2} \;\;=\;\; \frac{2x-y}{\sqrt{5}}$

. . . . . . . . . . . .$\displaystyle (x-5)^2 + (y-2)^2 \;\;=\;\; \frac{(2x-y)^2}{5}$

. . . . . .$\displaystyle x^2 - 10x + 25 + y^2 - 4y + 4 \;\;=\;\; \frac{4x^2 - 4xy + y^2}{5}$

. . $\displaystyle 5x^2 - 50x + 125 + 5y^2 - 20y + 20 \;\;=\;\; 4x^2 - 4xy + y^2$

. . . . . . $\displaystyle x^2 + 4xy -50x + 4y^2 - 20y + 145 \;\;=\;\;0$

Edit: corrected omitted term.
.

 

[galactus]

As usual, very nice Soroban. Simplistic and clever.

When I first encountered this, I started rotating and all that complicated mess.

 

[galactus]

Hey Soroban. I went ahead and graphed said parabola and y=2x.

See how the focus looks a little off center from where it looks like it should be?.

Wonder why?. Your assessment appears to be perfectly valid. I even thought along those lines after I

got away from the rotation mess. But, I did not finish and wanted another point of view.

Anyway, take a look. See what I mean?.

 

[Deleted member 4993]

Hey Soroban. I went ahead and graphed said parabola and y=2x.

See how the focus looks a little off center from where it looks like it should be?.

Wonder why?.

I think that is because the grid is not square.

Your assessment appears to be perfectly valid. I even thought along those lines after I

got away from the rotation mess. But, I did not finish and wanted another point of view.

Anyway, take a look. See what I mean?.

 

[galactus]

Yes, of course. Thanks SK. I am out of it today. :lol:

I adjusted the grid and reposted. It looks better now.

Thanks for pointing out my obvious brain fart/oversight, SK.