# What is the equation in standard form of the parabola with a focus at (1,5) and a directrix of y= 7?

$y = - \frac{1}{4} \cdot {x}^{2} + \frac{1}{2} \cdot x + \frac{23}{6}$
Focus is at (1,5) and directrix is y=7. So the distance between focus and directrix is $7 - 5 = 2 u n i t s$ Vertex is at the mid point between Focus and Directrix. So vertex co-ordinate is (1,6) . The parabola opens down as focus is below the Vertex. We know the equation of parabola is $y = a \cdot {\left(x - h\right)}^{2} + k$ where (h,k) is the vertex. Thus the Equation becomes $y = a \cdot {\left(x - 1\right)}^{2} + 6$ now $a = \frac{1}{4} \cdot c$where c is the distance between vertex and directrix; which is here equal to 1 so $a = - \frac{1}{4} \cdot 1 = - \frac{1}{4}$ (negative sign is as the parabola opens down) So the equation becomes $y = - \frac{1}{4} \cdot {\left(x - 1\right)}^{2} + 6 \mathmr{and} y = - \frac{1}{4} \cdot {x}^{2} + \frac{1}{2} \cdot x + \frac{23}{6}$graph{-1/4x^2+1/2x+23/6 [-10, 10, -5, 5]} [ans]