# What is the equation in standard form of the parabola with a focus at (1,4) and a directrix of y= 3?

##### 1 Answer
Jun 1, 2018

The equation of parabola is $y = \frac{1}{2} {\left(x - 1\right)}^{2} + 3.5$

#### Explanation:

Focus is at $\left(1 , 4\right)$and directrix is $y = 3$. Vertex is at midway

between focus and directrix. Therefore vertex is at $\left(1 , \frac{4 + 3}{2}\right)$

or at $\left(1 , 3.5\right)$ . The vertex form of equation of parabola is

y=a(x-h)^2+k ; (h.k) ; being vertex. $h = 1 \mathmr{and} k = 3.5$

So the equation of parabola is $y = a {\left(x - 1\right)}^{2} + 3.5$. Distance of

vertex from directrix is $d = 3.5 - 3 = 0.5$, we know $d = \frac{1}{4 | a |}$

$\therefore 0.5 = \frac{1}{4 | a |} \mathmr{and} | a | = \frac{1}{0.5 \cdot 4} = \frac{1}{2}$. Here the directrix is

below the vertex , so parabola opens upward and $a$ is positive.

$\therefore a = \frac{1}{2}$ . The equation of parabola is $y = \frac{1}{2} {\left(x - 1\right)}^{2} + 3.5$

graph{0.5(x-1)^2+3.5 [-20, 20, -10, 10]} [Ans]