Find the vertex and focus of the parabola below.
$$y ^ { 2 } + 10 y - 8 x + 30 = 0$$

A)
Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 11 } { 8 } , 5 \right)$

B)
Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 21 } { 8 } , - 5 \right)$

C)
Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 69 } { 8 } , - 5 \right)$

D)
Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , 5 \right)$

E)
Vertex: $\left( - \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , - 5 \right)$

Question

Find the vertex and focus of the parabola below.
$$y ^ { 2 } + 10 y - 8 x + 30 = 0$$

A)
Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 11 } { 8 } , 5 \right)$

B)
Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 21 } { 8 } , - 5 \right)$

C)
Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 69 } { 8 } , - 5 \right)$

D)
Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , 5 \right)$

E)
Vertex: $\left( - \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , - 5 \right)$

Quizplus · Accepted Answer

To find the vertex and focus of the parabola, first, rewrite the equation in a standard form. Completing the square for $y^2 + 10y$ gives $(y+5)^2$, so we rewrite the equation as $(y+5)^2 = 8x - 30$. Then, we further simplify to get $(y+5)^2 = 8(x - \frac{15}{4})$. This shows the parabola opens to the right, with the vertex at $(- \frac{15}{4}, -5)$ or $(\frac{5}{8}, -5)$ after simplifying. The focus is a distance $\frac{1}{4a}$ from the vertex along the axis of symmetry, where $a = 2$ (from the standard form $4a(x-h) = (y-k)^2$), so the focus is $\frac{1}{8}$ units to the right of the vertex, making it at $(\frac{5}{8} + \frac{1}{8}, -5)$ or $(\frac{21}{8}, -5)$.

Find the Vertex and Focus of the Parabola Below y2+10y−8x+30=0y ^ { 2 } + 10 y - 8 x + 30 = 0y2+10y−8x+30=0

Find the Vertex and Focus of the Parabola Below $y ^ { 2 } + 10 y - 8 x + 30 = 0$