Find equation of parabola given focus and directrix

Q: How do you find standard form when given focus and Directrix?

The standard form is (x - h)2 = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p. If the parabola is rotated so that its vertex is (h,k) and its axis of symmetry is parallel to the x-axis, it has an equation of (y - k)2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p.

This calculator will find either the equation of the parabola from the given parameters or the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length (distance), eccentricity, x-intercepts, y-intercepts, domain, and range of the entered parabola. Also, it will graph the parabola. Steps are available.

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How do you find standard form when given focus and Directrix?
What is parabola using focus and Directrix?

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Find the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length, eccentricity, x-intercepts, y-intercepts, domain, and range of the parabola $$$y = \left(x - 2\right)^{2} + 5$$$.

Solution

The equation of a parabola is $$$y = \frac{1}{4 \left(f - k\right)} \left(x - h\right)^{2} + k$$$, where $$$\left(h, k\right)$$$ is the vertex and $$$\left(h, f\right)$$$ is the focus.

Our parabola in this form is $$$y = \frac{1}{4 \left(\frac{21}{4} - 5\right)} \left(x - 2\right)^{2} + 5$$$.

Thus, $$$h = 2$$$, $$$k = 5$$$, $$$f = \frac{21}{4}$$$.

The standard form is $$$y = x^{2} - 4 x + 9$$$.

The general form is $$$x^{2} - 4 x - y + 9 = 0$$$.

The vertex form is $$$y = \left(x - 2\right)^{2} + 5$$$.

The directrix is $$$y = d$$$.

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: $$$5 - \frac{21}{4} = d - 5$$$.

Thus, the directrix is $$$y = \frac{19}{4}$$$.

The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: $$$x = 2$$$.

The focal length is the distance between the focus and the vertex: $$$\frac{1}{4}$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{1}{2}$$$.

The latus rectum is parallel to the directrix and passes through the focus: $$$y = \frac{21}{4}$$$.

The endpoints of the latus rectum can be found by solving the system $$$\begin{cases} x^{2} - 4 x - y + 9 = 0 \\ y = \frac{21}{4} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the latus rectum are $$$\left(\frac{3}{2}, \frac{21}{4}\right)$$$, $$$\left(\frac{5}{2}, \frac{21}{4}\right)$$$.

The length of the latus rectum is four times the distance between the vertex and the focus: $$$1$$$.

The eccentricity of a parabola is always $$$1$$$.

The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).

Since there are no real solutions, there are no x-intercepts.

The y-intercepts can be found by setting $$$x = 0$$$ in the equation and solving for $$$y$$$: (for steps, see intercepts calculator).

y-intercept: $$$\left(0, 9\right)$$$.

Answer

Standard form: $$$y = x^{2} - 4 x + 9$$$A.

General form: $$$x^{2} - 4 x - y + 9 = 0$$$A.

Vertex form: $$$y = \left(x - 2\right)^{2} + 5$$$A.

Focus-directrix form: $$$\left(x - 2\right)^{2} + \left(y - \frac{21}{4}\right)^{2} = \left(y - \frac{19}{4}\right)^{2}$$$A.

Graph: see the graphing calculator.

Vertex: $$$\left(2, 5\right)$$$A.

Focus: $$$\left(2, \frac{21}{4}\right) = \left(2, 5.25\right)$$$A.

Directrix: $$$y = \frac{19}{4} = 4.75$$$A.

Axis of symmetry: $$$x = 2$$$A.

Latus rectum: $$$y = \frac{21}{4} = 5.25$$$A.

Endpoints of the latus rectum: $$$\left(\frac{3}{2}, \frac{21}{4}\right) = \left(1.5, 5.25\right)$$$, $$$\left(\frac{5}{2}, \frac{21}{4}\right) = \left(2.5, 5.25\right)$$$A.

Length of the latus rectum: $$$1$$$A.

Focal parameter: $$$\frac{1}{2} = 0.5$$$A.

Focal length: $$$\frac{1}{4} = 0.25$$$A.

Eccentricity: $$$1$$$A.

x-intercepts: no x-intercepts.

y-intercept: $$$\left(0, 9\right)$$$A.

Domain: $$$\left(-\infty, \infty\right)$$$A.

Range: $$$\left[5, \infty\right)$$$A.

How do you find standard form when given focus and Directrix?

The standard form is (x - h)² = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p. If the parabola is rotated so that its vertex is (h,k) and its axis of symmetry is parallel to the x-axis, it has an equation of (y - k)² = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p.

What is parabola using focus and Directrix?

In mathematics, a parabola is the locus of a point that moves in a plane where its distance from a fixed point known as the focus is always equal to the distance from a fixed straight line known as directrix in the same plane.