Find the vertex, the focus, and the directrix of the parabola.
-$$2 x ^ { 2 } + 20 x - y + 53 = 0$$
A) Vertex: $( - 5,0 )$; Focus: $( - 5,5 )$; Directrix: $x = 1$
B) Vertex: $\left( - 5 , - \frac { 24 } { 17 } \right)$; Focus: $\left( - 5 , \frac { 23 } { 8 } \right) ;$ Directrix: $x = \frac { 25 } { 8 }$
C) Vertex: $( - 5,3 )$; Focus: $( - 5,11 )$; Directrix: $x = 5$
D) Vertex: $( - 5,3 )$; Focus: $\left( - 5 , \frac { 25 } { 8 } \right)$; Directrix: $y = \frac { 23 } { 8 }$

Question

Quizplus · Accepted Answer

To find the vertex, focus, and directrix of the parabola given by $2x^2 + 20x - y + 53 = 0$, we first complete the square for the $x$-terms and rewrite the equation in standard form. Completing the square involves adding and subtracting $(\frac{b}{2a})^2$, where $a = 2$ and $b = 20$, inside the equation. This gives us the vertex form of a parabola, $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex. After rearranging and completing the square, we find the vertex is $(-5, 3)$. The focus and directrix of a parabola are determined by the distance $1/(4a)$ from the vertex, where $a$ is the coefficient of the $(x-h)^2$ term. Given $a = 2$ in the standard form, the focus is $\frac{1}{4a}$ units away from the vertex along the axis of symmetry, and the directrix is the same distance in the opposite direction. Calculating these values gives us the focus at $\left( - 5 , \frac { 25 } { 8 } \right)$ and the directrix at $y = \frac { 23 } { 8 }$.

Find the Vertex, the Focus, and the Directrix of the Parabola