Question

The equation of the parabola with focus (0,0) and directrix x+y=413
- 2xy + 8x + 8y-16 = 0 (b) y - 2xy + 8x + 8y
(c) x² + y² + 8x + 8y-16 = 0 (d) 12 y2 + 8x + 8y -16 = 0​

Answer 1

Therefore the equation of the parabola is

$x^2+y^2-x-y+413=0$

Step-by-step explanation:

Parabola: Any point of a parabola is at an equal distance from the focus and the directrix.

Let (a,b) be a point on the the parabola.

The distance between two point (x₁,y₁) and (x₂,y₂) is

$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The distance between (a,b) and the focus(0,0) is$\sqrt{(a-0)^2+(b-0)^2}$

                                                                             =   $\sqrt{a^2+b^2}$

The distance between a plane ax+by+c=0 and (p,q)is

$=|\frac{ap+bq+c}{\sqrt{a^2+b^2} } |$

So, the distance between x+y=413 and (a,b) is

$|\frac{a+b-413}{\sqrt {a^2+b^2}} |$

Therefore,

$\sqrt{a^2+b^2}=|\frac{a+b-413}{\sqrt {a^2+b^2}} |$

$\Rightarrow a^2+b^2}={a+b-413}$

Therefore the locus of (a,b) is

$x^2+y^2=x+y-413$                 [ putting a=x and b=y]

Therefore the equation of the parabola is

$x^2+y^2-x-y+413=0$