Given that the equation of the parabola is 5y^2 + 24x = 0.
Find (1)The Axis and vertex of the parabola
(ii)The focus and the directrix
(iii)The distance from the directrix to the focus
24x = -5y^2.
X = (-5/24)Y^2.
Solution set: (0.0).
To find the axis and vertex of the parabola, we need to rewrite the equation in the standard form of a parabola, which is (y-k)^2 = 4a(x-h), where (h, k) is the vertex and a is the distance between the vertex and the focus/directrix.
1) The Axis and vertex of the parabola:
To find the vertex, rearrange the equation:
5y^2 = -24x
Divide both sides by 5:
y^2 = -24/5 x
Now, we can compare this equation with the standard form (y-k)^2 = 4a(x-h):
The coefficient of x in our equation is -24/5, so h = -24/5.
The vertex of the parabola is given by (h, k), so the vertex is (-24/5, 0).
The axis of the parabola is a vertical line passing through the vertex. So, the equation of the axis is x = -24/5.
2) The focus and the directrix:
To find the focus and directrix, we need the value of 'a'. In the standard form equation (y-k)^2 = 4a(x-h), 'a' is the distance between the vertex and the focus/directrix.
Comparing this standard form equation with our given equation (y^2 = -24/5 x), we can see that the coefficient of x is -24/5, which means a = -24/5.
Now, we can find the focus and the directrix using the formulas:
Focus(F): (h+a, k)
Directrix(D): x = h-a
From (1), we know that the vertex is (-24/5, 0). Substitute the values of h, k, and a:
Focus(F): (-24/5 + (-24/5), 0) = (-48/5, 0)
Directrix(D): x = -24/5 - (-24/5) = -24/5 + 24/5 = 0
So, the focus is (-48/5, 0), and the directrix is x = 0.
3) The distance from the directrix to the focus:
The distance from the directrix to the focus is equal to 'a', which is the distance between the vertex and the focus or the vertex and the directrix. From our calculations, we found that a = -24/5.
Therefore, the distance from the directrix to the focus is |-24/5|, which is equal to 24/5.
To find the axis and vertex of the parabola:
Step 1: Rewrite the given equation in standard form: y^2 = (-24/5)x.
Step 2: Compare the equation with the standard form y^2 = 4px, where p is the distance from the vertex to the focus.
Step 3: Therefore, p = -24/5. Since p is negative, the parabola opens to the left.
Step 4: The axis of the parabola is a vertical line that passes through the vertex. In this case, the axis is the y-axis because the equation does not contain any x term.
Step 5: The vertex is the point (0,0) since the vertex is always on the axis of the parabola.
To find the focus and the directrix:
Step 1: Use the formula x = -p to find the x-coordinate of the focus.
x = -(-24/5) = 24/5.
Step 2: The focus is the point (24/5, 0) since the parabola opens to the left and the focus lies on the positive x-axis.
Step 3: The directrix is a vertical line that is p distance away from the vertex on the opposite side of the parabola. In this case, the directrix is the line x = -24/5.
To find the distance from the directrix to the focus:
Step 1: The distance from the directrix to the focus is equal to |p|.
|p| = |-24/5| = 24/5.
Therefore, for the given equation of the parabola:
(i) The axis is the y-axis, and the vertex is (0,0).
(ii) The focus is (24/5, 0), and the directrix is x = -24/5.
(iii) The distance from the directrix to the focus is 24/5.