How do you find an equation of a parabola given vertex (7,0) and focus (0,0)?

Dec 19, 2017

${y}^{2} = - 28 x + 196$

Explanation:

As the vertex is $\left(7 , 0\right)$ and focus is $\left(0 , 0\right)$, the line joining them is axis of symmetry i.e. $y = 0$

Directrix is perpendicular to axis of symmetry and hence its equation is of the type $x = k$

Further as vertex is equidistant from focus and directrix and is also between the two, we must have directrix as $x = 14$

Now parabola is locus of a point, say $\left(x , y\right)$, which moves so that its distance from focus and directrix is always equal,

as distance of $\left(x , y\right)$ from $\left(0 , 0\right)$ is $\sqrt{{x}^{2} + {y}^{2}}$

and distance of $\left(x , y\right)$ from $x = 14$ is $| x - 14 |$,

equation of parabola is $\sqrt{{x}^{2} + {y}^{2}} = | x - 14 |$

or ${x}^{2} + {y}^{2} = {x}^{2} - 28 x + 196$

or ${y}^{2} = - 28 x + 196$

graph{(y^2+28x-196)(x^2+y^2-0.2)((x-7)^2+y^2-0.2)(x-14)=0 [-33.4, 46.6, -20.16, 19.84]}