Question

The equation of the parabola with focus (0,0) and directrix x+y=4 is

• A. $x^2+y^2-2xy+8x+8y-16=0$
• B. $x^2+y^2-2xy+8x+8y=0$
• C. $x^2+y^2+8x+8y-16=0$
• D. $x^2-y^2+8x+8y-16=0$

Answer 1

The correct option is A

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

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

Draw PM perpendicular to x+y=4

Then,we have: SP=PM

$\Rightarrow S{P}^2=P{M}^2$

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

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

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

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