The equation has a vertex at (0, 5) and focus of (three-halves, 5) is,
- 6 x + y²- 10 y + 25 = 0
−6x+y² −10y+25=0
What is a parabola?
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point and a line.
Given
The equation of a parabola is x = [tex]\frac{1}{4 \left(f - h\right)} \left(y - k\right)^{2} + h[/tex], where (h,k) is the vertex and (f,k) is the focus.
Thus, h = 0 k = 5, f = 3/2
The standard form is x = [tex]\frac{y^{2}}{6} - \frac{5 y}{3} + \frac{25}{6}x[/tex]
The general form is - 6 x + y² - 10 y + 25 = 0
The vertex form is x = [tex]\frac{\left(y - 5\right)^{2}}{6}[/tex]
The directrix is x = d
To find d, use the fact that the distance from the focus to the vertex is the same as the distance from the vertex to the directrix: [tex]0 - \frac{3}{2} = d - 0[/tex]
Thus, the directrix is x = - 3/2
The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: y = 5
The focal length is the distance between the focus and the vertex: 3/2
The length of the latus rectum is four times the distance between the vertex and the focus: 6.
The eccentricity of a parabola is always 1.
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