Find the vertex, focus, axis, directrix and latus rectum of the parabola x^2+ y = 6x - 14
To find the vertex, focus, axis, directrix, and latus rectum of the parabola given by the equation x^2 + y = 6x - 14, we need to rewrite the equation in the standard form of a parabola (y = a(x-h)^2 + k).
First, we move all terms to the left side:
x^2 - 6x + y + 14 = 0
Next, we complete the square to rewrite the equation in the standard form:
(x^2 - 6x + 9) + y + 14 - 9 = 9
(x - 3)^2 + y + 5 = 9
By comparing this equation with the standard form, we can determine the vertex, axis of symmetry, and shift amount (h and k values):
Vertex: The vertex of the parabola is the point (h, k) in the standard form equation. In this case, the vertex is (3, -5).
Axis of symmetry: The axis of symmetry is a vertical line passing through the vertex. The equation of the axis of symmetry is x = h. So, for this parabola, the axis of symmetry is x = 3.
Now, to find the value of 'a', we compare the standard form equation with the given equation. The coefficient of the squared term in the standard form is 'a'.
(a = 1 in this case since y is not multiplied by any coefficient)
Since a = 1, the parabola opens upward.
Now, we can find the focus and directrix:
Focus: The focus point (F) is a point on the axis of symmetry that is equidistant from the vertex and the directrix. For a parabola opening upward, the focus point is (h, k+a), where 'a' is the coefficient of the squared term. So, for this parabola, the focus is (3, -5+1) = (3, -4).
Directrix: The directrix is a horizontal line that is equidistant from the focus and the vertex. For a parabola opening upward, the equation of the directrix is y = k - a. So, for this parabola, the directrix is y = -5-1 = -6.
Latus Rectum: The latus rectum is the line segment passing through the focus, perpendicular to the axis of symmetry, and it intersects the parabola at two points. The length of the latus rectum is given by 4a, where 'a' is the coefficient of the squared term. So, for this parabola, the length of the latus rectum is 4(1) = 4.