# How do you find the equation of a parabola with directrix y=-1 and focus F(0,0)?

Mar 8, 2017

The equation of a parabola is ${x}^{2} - 2 y - 1 = 0$

#### Explanation:

Parabola is the locus of a point, which moves so that its distance from a line, called directix, and a point, called focus, are equal.

We know that the distance between two points $\left({x}_{1} , {y}_{1}\right)$ and x_2,y_2) is given by $\sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$ and

the distance between point $\left({x}_{1} , {y}_{1}\right)$ and line $a x + b y + c = 0$ is |ax_1+by_1+c|/(sqrt(a^2+b^2).

Now distance of a point $\left(x , y\right)$ on parabola from focus at $\left(0 , 0\right)$ is $\sqrt{{x}^{2} + {y}^{2}}$

and its distance from directrix $y = - 1$ or $y + 1 = 0$ is $| y + 1 \frac{|}{\sqrt{{0}^{2} + {1}^{2}}} = | y + 1 |$

Hence, equation of parabola would be

$\sqrt{{x}^{2} + {y}^{2}} = | y + 1 |$ or

${x}^{2} + {y}^{2} = {\left(y + 1\right)}^{2}$ or

${x}^{2} + {y}^{2} = {y}^{2} + 2 y + 1$ or

${x}^{2} - 2 y - 1 = 0$

graph{x^2-2y-1=0 [-2.592, 2.408, -1.07, 1.43]}