Question

Write the equation of the parabola with focus (0,0) and directrix x+y-4=0.

Answer 1

Let P(x,y) be any point on the parabola whose focus is S(0,0) and the directrix x+y-4=0

Draw PM perpendicular from P(x,y) on the directrix

x+y-4=0

Then by definition, SP=PM

$\Rightarrow S{P}^2=P{M}^2\Rightarrow (x-0{)}^2+(y-0{)}^2={\left|\frac{x+y-4}{\sqrt{1^2+1^2}}\right|}^2$

$\Rightarrow x^2+y^2=\frac{(x+y-4{)}^2}{(\sqrt{2}{)}^2}$

$\Rightarrow 2x^2+2y^2=x^2+y^2+(-4{)}^2+2xy+2\times y\times (-4)+2\times (-4)\times x$

$\Rightarrow 2x^2+2y^2=x^2+y^2+16+2xy-8y-8x$

$\Rightarrow 2x^2-x^2+2y^2-y^2-2xy+8x+8y-16=0$

$\Rightarrow x^2+y^2-2xy+8x+8y-16=0$