Find the vertex focus directrix and axis of symmetry of each parabola y²–8x–6y–3=0
y²–8x–6y–3=0
looks like a horizontal parabola, let's complete the square
y^2 - 6y + 9 = 8x+3 + 9
(y - 3)^2 = 8x + 12
= 8(x + 3/2)
from that version you should be able to pick out the vertex as (-3/2 , 3)
recall that in y^2 = 4px the focus is (p,0) and the directrix is y = -p
So from you equation, 4p = 8, then p = 2
the focus is(2-3/2, 3), or (-1/2 , 3)
and the directrix is ........
https://www.wolframalpha.com/input/?i=y%C2%B2%E2%80%938x%E2%80%936y%E2%80%933%3D0
To find the vertex, focus, directrix, and axis of symmetry for the parabola given by the equation y²–8x–6y–3=0, we can start by rearranging the equation into the standard form of a parabola, which is (x-h)² = 4p(y-k).
1. Rearrange the equation:
y²–8x–6y–3=0
y² – 6y – 8x – 3 = 0
2. Complete the square for the y terms:
(y² – 6y) – 8x – 3 = 0
(y² – 6y + 9) – 9 – 8x – 3 = 0
(y - 3)² - 8x + 6 = 0
3. Move any constants to the right side:
(y - 3)² = 8x - 6
The equation is now in the form (x-h)² = 4p(y-k), where (h, k) is the vertex.
Comparing this form to our equation:
(x - h)² = 4p(y - k)
We can see that h = 0 and k = 3.
Therefore, the vertex of the parabola is (0, 3).
To find the axis of symmetry, we look at the x-coordinate of the vertex. In this case, the axis of symmetry is x = 0.
Now, let's find the value of p to determine the focus and directrix.
4. Compare the equation to the standard form:
4p = 8
p = 2
The value of p tells us how far the focus is from the vertex, as well as the distance between the vertex and the directrix.
The focus is (h + p, k) = (0 + 2, 3) = (2, 3).
The directrix is given by the equation x = (h - p) = (0 - 2) = -2.
In summary:
- Vertex: (0, 3)
- Focus: (2, 3)
- Directrix: x = -2
- Axis of symmetry: x = 0